Chapter Algebra of matrices

ALGEBRA OF MATRICES

EXERCISE 5.1

1.If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

solution:

Order of matrix=number of rows ​\( \times \)​number of column

or

Total number of elements=number of rows ​\( \times \)​number of column

Thus, to find all possible order of a matrix having 8 elements we have to find all possible ordered pairs whose product is 8.

Possible order pairs are

(1\( \times \)8) ; (8\( \times \)1) ; (2​\( \times \)​4) ; (4\( \times \)2) ;

AND

The possible orders of a matrix with 5 total elements are

(1 \( \times \) 5) ; (5 \( \times \) 1)

2.If 

\[ A=[​a_{ij}​]=\begin{bmatrix}2&3&-5\\1&4&~~~9\\0&7&-2\end{bmatrix}~and~B=[​b_{ij}]= \begin{bmatrix}~~~2&-1\\-3&~~~4\\~~~1&~~~2\end{bmatrix} \]

then find 

(i)​\( a_{22}+b_{21} \) ​(ii)​\( a_{11}b_{11}+a_{22}b_{22} \)

solution:

(i)\( a_{22}+b_{21} \)

where,

\( a_{22}=4 \)​and ​\( b_{21}=-3 \)

\( =a_{22}+b_{21} \)

\( =4+(-3) \)

\( =1 \)

(ii)\( a_{11}b_{11}+a_{22}b_{22} \)

where,

\( a_{11}=2~;~b_{11}=2~; \)

\( \\a_{22}=4~;~b_{22}=4 \)

\( =a_{11}b_{11}+a_{22}b_{22} \)

\( \\=(2)(2)+(4)(4) \)

\( =4+16 \)

\( =20 \)

3.Let A be a matrix of order 3​\( \times \)​4.If ​\( R_1 \)​denotes the first row of A and ​\( C_2 \)​denotes second column, then determine the orders of matrices ​\( R_1 \)​ and ​\( C_2 \)​.

4.Construct a 2​\( \times \)​3 matrix whose elements ​\( a_{ij} \)​are given by:

(i)​\( a_{ij}=i.j \)

(ii)​\( a_{ij}=2i-j \)

(iii)​\( a_{ij}=i+j \)

(iv)​\( a_{ij}=\dfrac{(i+j)^2}{2} \)

5.Construct a 2 ​\( \times \)​2 matrix whose elements ​\( a_{ij} \)​are given by:

(i)\( a_{ij}=\dfrac{(i+j)^2}{2} \)

(ii) \( a_{ij}=\dfrac{(i-j)^2}{2} \)

(iii)\( a_{ij}=\dfrac{(i-2j)^2}{2} \)​​

(iv)\( a_{ij}=\dfrac{(2i+j)^2}{2} \)

(v)\( a_{ij}=\dfrac{\lvert{2i-3j}\rvert}{2} \)

(vi)\( a_{ij}=\dfrac{\lvert{-3i+j}\rvert}{2} \)

6.Construct a 3 ​\( \times \)​ 4 matrix A=[​\( a_{ij} \)​] whose elements ​\( a_{ij} \)​are given by:

(i)\( a_{ij}=i+j \)

(ii)\( a_{ij}=i-j \)

(iii)\( a_{ij}=2i \)

(iv)\( a_{ij}=j \)

(v)\( a_{ij}=\dfrac{1}{2}~\lvert{-3i+j}\rvert \)

7.Construct a matrix 4​\( \times \)​3 matrix whose elements are 

(i)​\( a_{ij}=2i+\dfrac{i}{j} \)

(ii)​\( a_{ij}=\dfrac{i-j}{i+j} \)

(iii)​\( a_{ij}=i \)

8.Find x,y,a and b if ​

\[ \begin{bmatrix}3x+4y&2&x-2y\\a+b&2a-b&-1\end{bmatrix}=\begin{bmatrix}2&2&4\\5&-5&-1\end{bmatrix} \]

9.Find x,y,a and b if 

(i)

\[ \begin{bmatrix}2x-3y&a-b&3\\1&x+4y&3a+4b\end{bmatrix}=\begin{bmatrix}1&-2&3\\1&~~~6&29\end{bmatrix} \]

10.Find the value of a,b,c and d from the following equations:

\( \begin{bmatrix}2a+b&a-2b\\5c-d&4c+3d\end{bmatrix}=\begin{bmatrix}4&-3\\11&24\end{bmatrix} \)

11.For what values of x and y are the following matrices equal ?

\( A=\begin{bmatrix}2x+1&2y\\0&y^2-5y\end{bmatrix} \)​,

\( B=\begin{bmatrix}x+3&y^2+2\\0&-6\end{bmatrix} \)

12.Find x,y and z so that A=B,where

\( A=\begin{bmatrix}x-2&3&2z\\18z&y+2&6z\end{bmatrix} \)​,

\( B=\begin{bmatrix}y&z&6\\6y&x&2y\end{bmatrix} \)

13.If \( \begin{bmatrix}x&3x-y\\2x+z&3y-w\end{bmatrix}=\begin{bmatrix}3&2\\4&7\end{bmatrix} \)​,find x,y,z,w.

14.If ​\( \begin{bmatrix}x-y&z\\2x+y&w\end{bmatrix}=\begin{bmatrix}-1&4\\~~~0&5\end{bmatrix} \)​,find x,y,z,w.

15.Find the values of x and y if

\[ \begin{bmatrix}x+10&y^2+2y\\0&-4\end{bmatrix}=\begin{bmatrix}3x+4&3\\0&y^2-5y\end{bmatrix} \]

16.If

\[ \begin{bmatrix}x+3&z+4&2y-7\\4x+6&a-1&0\\b-3&3b&z+2c\end{bmatrix}=\begin{bmatrix}0&~~~6&3y-2\\2x&-3&2c-2\\2b+4&-21&0\end{bmatrix} \]

Obtain the value of a,b,c,x,y and z.

17.Give an example of
(i)a row matrix which is also a column matrix,
(ii)a diagonal matrix which is not scalar,
(iii)a triangular matrix

18.The sale figure of two car dealers during January 2007 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars,while dealer B sold 7 deluxe, 2 premium and 3 standard cars.Total sales over the 2 month periods of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars.In the same 2 month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars.Write 2​\( \times \)​3 matrices summarizing sales data for January and 2 month periods for each dealer.

EXERCISE 5.2

1.Compute the following sums:

(i)​\( \begin{bmatrix}3&-2\\1&~~~4\end{bmatrix}+\begin{bmatrix}-2&4\\~~~1&3\end{bmatrix} \)

(ii)\( \begin{bmatrix}~~~2&1&3\\~~~0&3&5\\-1&2&5\end{bmatrix}+\begin{bmatrix}1&-2&3\\2&~~~6&1\\0&-3&1\end{bmatrix} \)

2.Let A=​\( \begin{bmatrix}2&4\\3&2\end{bmatrix} \)​, B=​\( \begin{bmatrix}~~~1&3\\-2&5\end{bmatrix} \)​ and C=​\( \begin{bmatrix}-2&5\\~~~3&4\end{bmatrix} \)​.Find each of the following:

(i)2A-3B

(ii)B-4C

(iii)3A-C

(iv)3A-2B+3C

3.If A=​\( \begin{bmatrix}2&3\\5&7\end{bmatrix} \)​,B=​\( \begin{bmatrix}-1&0&2\\~~~3&4&1\end{bmatrix} \)​, C=​\( \begin{bmatrix}-1&2&3\\~~~2&1&0\end{bmatrix} \),find :

(i)A+B and B+C

(ii)2B+3A and 3C-4B

4.Let A=​\( \begin{bmatrix}-1&0&2\\~~~3&1&4\end{bmatrix} \)​, B=​\( \begin{bmatrix}0&-2&5\\1&-3&1\end{bmatrix} \)​ and C=​\( \begin{bmatrix}1&-5&2\\6&~~~0&-4\end{bmatrix} \)​. Compute 2A-3B+4C.

5.If A=diag(2 -5 9), B=diag(1 1 -4) and C=diag(-6 3 4) , find:

(i) A-2B

(ii) B+C-2A

(iii) 2A+3B-5C

6. Given the matrices

A=​\( \begin{bmatrix}2&~~~1&1\\3&-1&0\\0&~~~2&4\end{bmatrix} \)​, B=​\( \begin{bmatrix}9&7&-1\\3&5&~~~4\\2&1&~~~6\end{bmatrix} \)​ and C=​\( \begin{bmatrix}2&-4&3\\1&-1&0\\9&~~~4&5\end{bmatrix} \)

Verify that (A+B)+C=A+(B+C).

7.Find the matrices X and Y, if X+Y=​\( \begin{bmatrix}5&2\\0&9\end{bmatrix} \)​ and X-Y=​\( \begin{bmatrix}3&~~~6\\0&-1\end{bmatrix} \)

8.Find X if Y=​\( \begin{bmatrix}3&2\\1&4\end{bmatrix} \)​ and 2X+Y=​\( \begin{bmatrix}~~~1&0\\-3&2\end{bmatrix} \)

9.Find the matrices X and Y , if 2X-Y=​\( \begin{bmatrix}~~~6&-6&0\\-4&~~~2&1\end{bmatrix} \)​ and X+2Y=​\( \begin{bmatrix}~~~3&2&~~~5\\-2&1&-7\end{bmatrix} \)

10.If X-Y=​\( \begin{bmatrix}1&1&1\\1&1&0\\1&0&0\end{bmatrix} \)​ and X+Y=​\( \begin{bmatrix}~~~3&5&1\\-1&1&4\\~~~11&8&0\end{bmatrix} \)​ , find X and Y.

11.Find the matrix A, if ​\( \begin{bmatrix}1&2&-1\\0&4&~~~9\end{bmatrix} \)​+A=​\( \begin{bmatrix}~~~9&-1&4\\-2&~~~1&3\end{bmatrix} \)

12.If A=​\( \begin{bmatrix}9&1\\7&8\end{bmatrix} \)​ , B=​\( \begin{bmatrix}1&5\\7&12\end{bmatrix} \)​, find matrix C such that 5A+3B+2C is a null matrix.

13.If A=​\( \begin{bmatrix}~~~2&-2\\~~~4&~~~2\\-5&~~~1\end{bmatrix} \)​, B=​\( \begin{bmatrix}8&~~~0\\4&-2\\3&~~~6\end{bmatrix} \)​, find matrix X such that 2A+3X=5B.

14.If A=​\( \begin{bmatrix}1&-3&2\\2&~~~0&2\end{bmatrix} \)​ and B=​\( \begin{bmatrix}2&-1&-1\\1&~~~0&-1\end{bmatrix} \)​, find the matrix C such that A+B+C is zero matrix.

15.Find x, y satisfying the matrix equations:

(i)

\[ \begin{bmatrix}x-y&2&-2\\4&x&~~~6\end{bmatrix}+\begin{bmatrix}3&-2&~~~2\\1&~~~0&-1\end{bmatrix}=\begin{bmatrix}6&0&0\\5&2x+y&5\end{bmatrix} \]

(ii)

\[ \begin{bmatrix}x&y+2&z-3\end{bmatrix}+\begin{bmatrix}y&4&5\end{bmatrix}=\begin{bmatrix}4&9&12\end{bmatrix} \]

16.If ​

\[ 2\begin{bmatrix}3&4\\5&x\end{bmatrix}+\begin{bmatrix}1&y\\0&1\end{bmatrix}=\begin{bmatrix}7&0\\10&5\end{bmatrix} \]

​,find x and y.

17.Find the value of ​\( \lambda \)​, a non-zero scalar , if

\[ \lambda\begin{bmatrix}1&0&2\\3&4&5\end{bmatrix}+2\begin{bmatrix}~~~1&~~~2&3\\-1&-3&2\end{bmatrix}=\begin{bmatrix}4&4&10\\4&2&14\end{bmatrix} \]

18. (i) Find a matrix X such that 2A+B+X=O, where

A=​\( \begin{bmatrix}-1&2\\~~~3&4\end{bmatrix} \)​, B=​\( \begin{bmatrix}3&-2\\1&~~~5\end{bmatrix} \)

(ii) If A=​\( \begin{bmatrix}8&~~~0\\4&-2\\3&~~~6\end{bmatrix} \)​ and B=​\( \begin{bmatrix}~~~2&-2\\~~~4&~~~2\\-5&~~~1\end{bmatrix} \)​, then find the matrix X of order 3​\( \times \)​2 such that 2A+3X=5B.

19.Find x, y, z and t, if 

(i)

\[ 3\begin{bmatrix}x&y\\z&t\end{bmatrix}=\begin{bmatrix}~~~x&6\\-1&2t\end{bmatrix}+\begin{bmatrix}4&x+y\\z+t&3\end{bmatrix} \]

(ii)

\[ 2\begin{bmatrix}x&5\\7&y-3\end{bmatrix}+\begin{bmatrix}3&4\\1&2\end{bmatrix}=\begin{bmatrix}7&14\\15&14\end{bmatrix} \]

20.In a certain city there are 30 colleges. Each colleges has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix . Using scalar multiplication, find the total number of posts of each kind in all the colleges.

EXERCISE 5.3

1.Compute the indicated products:

(i) ​\( \begin{bmatrix}~~~a&b\\-b&a\end{bmatrix}\begin{bmatrix}a&-b\\b&a\end{bmatrix} \)

(ii) ​\( \begin{bmatrix}1&-2\\2&~~~3\end{bmatrix}\begin{bmatrix}~~~1&2&~~~3\\-3&2&-1\end{bmatrix} \)

(iii)\( \begin{bmatrix}2&3&4\\3&4&5\\4&5&6\end{bmatrix}\begin{bmatrix}1&-3&5\\0&~~~2&4\\3&~~~0&5\end{bmatrix} \)

2. Show that AB​\( \not= \)​BA in each of the following cases:

(i) A=​\( \begin{bmatrix}5&-1\\6&~~~7\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}2&1\\3&4\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}-1&~~~1&0\\~~~0&-1&1\\~~~2&~~~3&4\end{bmatrix} \)​ and B=​\( \begin{bmatrix}1&2&3\\0&1&0\\1&1&0\end{bmatrix} \)

(iii) A=​\( \begin{bmatrix}1&3&0\\1&1&0\\4&1&0\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}0&1&0\\1&0&0\\0&5&1\end{bmatrix} \)

3.Compare the products AB and BA whichever exists in each of the following cases:

(i) A=​\( \begin{bmatrix}1&-2\\2&~~~3\end{bmatrix} \)​ and B=​\( \begin{bmatrix}1&2&3\\2&3&1\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}~~~3&2\\-1&0\\-1&1\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}4&5&6\\0&1&2\end{bmatrix} \)

(iii) A=​\( \begin{bmatrix}1&-1&2&3\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}0\\1\\3\\2\end{bmatrix} \)

(iv)\( [a,b]\begin{bmatrix}c\\d\end{bmatrix}+\begin{bmatrix}a&b&c&d\end{bmatrix}\begin{bmatrix}a\\b\\c\\d\end{bmatrix} \)

4. Show that AB​\( \not \)​BA in each of the following cases:

(i) A=​\( \begin{bmatrix}1&~~~3&-1\\2&-1&-1\\3&~~~0&-1\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}-2&3&-1\\-1&2&-1\\-6&9&-4\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}~~~10&-4&-1\\-11&~~~5&0\\~~~9&-5&1\end{bmatrix} \)​ and B=​\( \begin{bmatrix}1&2&1\\3&4&2\\1&3&2\end{bmatrix} \)

5. Evaluate the following:

(i)

\[ \Bigg(\begin{bmatrix}~~~1&~~~3\\-1&-4\end{bmatrix}+\begin{bmatrix}~~~3&-2\\-1&~~~1\end{bmatrix}\Bigg)\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix} \]

(ii)

\[ \begin{bmatrix}1&2&3\end{bmatrix}\begin{bmatrix}1&0&2\\2&0&1\\0&1&2\end{bmatrix}\begin{bmatrix}2\\4\\6\end{bmatrix} \]

(iii)

\[ \begin{bmatrix}1&-1\\0&~~~2\\2&~~~3\end{bmatrix}\Bigg(\begin{bmatrix}1&0&2\\2&0&1\end{bmatrix}-\begin{bmatrix}0&1&2\\1&0&2\end{bmatrix}\Bigg) \]

6. If A=​\( \begin{bmatrix}1&0\\0&1\end{bmatrix} \)​, B=​\( \begin{bmatrix}1&~~~0\\0&-1\end{bmatrix} \)​ and C=​\( \begin{bmatrix}0&1\\1&0\end{bmatrix} \),then show that ​\( A^2=B^2=C^2=l^2 \)

7. If A=​\( \begin{bmatrix}2&-1\\3&~~~2\end{bmatrix} \)​ and B=​\( \begin{bmatrix}~~~0&4\\-1&7\end{bmatrix} \)​,find ​\( 3A^2-2B+I \)

8. If A=​\( \begin{bmatrix}~~4&2\\-1&1\end{bmatrix} \)​,prove that (A-2I)(A-3I)=0

9. If A=​\( \begin{bmatrix}1&1\\0&1\end{bmatrix} \)​,show that ​\( A^2=\begin{bmatrix}1&2\\0&1\end{bmatrix} \)​and ​\( A^3=\begin{bmatrix}1&3\\0&1\end{bmatrix} \)

10. If A=​\( \begin{bmatrix}~~~ab&~~~b^2\\-a^2&-ab\end{bmatrix} \)​,show that ​\( A^2=0 \)

11. If A=​\( \begin{bmatrix}~~~cos2\theta&sin2\theta\\-sin2\theta&cos2\theta\end{bmatrix} \)​, find ​\( A^2 \)

12. If A=​\( \begin{bmatrix}~~~2&-3&-5\\-1&~~~4&~~~5\\~~~1&-3&-4\end{bmatrix} \)​ and B=​\( \begin{bmatrix}-1&~~~3&~~~5\\~~~1&-3&-5\\-1&~~~3&~~~5\end{bmatrix} \)​show that Ab=BA=​\( O_{{3}\times{3}} \)

13. If A=​\( \begin{bmatrix}~~~0&~~~c&-b\\-c&~~~0&~~~a\\~~~b&-a&~~~0\end{bmatrix} \)​ and B=​\( \begin{bmatrix}a^2&ab&ac\\ab&b^2&bc\\ac&bc&c^2\end{bmatrix} \)​show that AB=BA=​\( O_{{3}\times{3}} \)

14. If A=​\( \begin{bmatrix}2&-3&-5\\-1&~~~4&~~~5\\1&-3&-4\end{bmatrix} \)​ and B=​\( \begin{bmatrix}~~~2&-2&~~~4\\-1&~~~3&~~~4\\~~~1&-2&-3\end{bmatrix} \)​, show that AB=A and BA=B.

15. Let A=​\( \begin{bmatrix}-1&~~~1&-1\\~~~3&-3&~~~3\\~~~5&~~~5&~~~5\end{bmatrix} \)​ and B=​\( \begin{bmatrix}~~~0&~~~4&~~~3\\~~~1&-3&-3\\-1&~~~4&~~~4\end{bmatrix} \)​, compute ​\( A^2-B^2 \)

16. For the following matrices verify the associativity of matrix multiplicationi.e. (AB)C=A(BC):

(i) A=\( \begin{bmatrix}~~~1&~~2&~~0\\-1&~~0&~~~1\end{bmatrix} \), B=​\( \begin{bmatrix}~~~1&0\\-1&2\\~~~0&3\end{bmatrix} \)​ and C=​\( \begin{bmatrix}~~~1\\-1\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}4&2&3\\1&1&2\\3&0&2\end{bmatrix} \)​, B=​\( \begin{bmatrix}1&-1&1\\0&~~~1&2\\2&-1&1\end{bmatrix} \)​ , C=​\( \begin{bmatrix}1&2&-1\\3&0&~~~1\\0&0&~~~1\end{bmatrix} \)

17. For the following matrices verify the distributivity of the matrix multiplication over matrix addition i.e. A(B+C)=AB+AC

(i) A=​\( \begin{bmatrix}1&-1\\0&~~~2\end{bmatrix} \)​ , B=​\( \begin{bmatrix}-1&0\\~~~2&1\end{bmatrix} \)​ , C=​\( \begin{bmatrix}0&~~~1\\1&-1\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}~~~2&-1\\~~~1&~~~1\\-1&~~~2\end{bmatrix} \)​ , B=​\( \begin{bmatrix}0&1\\1&1\end{bmatrix} \)​ , C=​\( \begin{bmatrix}1&-1\\0&~~~1\end{bmatrix} \)

18. If A=​\( \begin{bmatrix}~~~1&~~~0&-2\\~~~3&-1&~~~0\\-2&~~~1&~~~1\end{bmatrix} \)​ , B=​\( \begin{bmatrix}~~~0&5&-4\\-2&1&~~~3\\-1&0&~~~2\end{bmatrix} \)​ , C=​\( \begin{bmatrix}~~~1&5&2\\-1&1&0\\~~~0&-1&1\end{bmatrix} \)

19.Compute the following elements ​\( a_{43} \)​ and ​\( a_{22} \)​ of the matrix :

A=​\( \begin{bmatrix}0&1&0\\2&0&2\\0&3&2\\4&0&4\end{bmatrix}\begin{bmatrix}~~~2&-1\\-3&~~~2\\~~~4&~~~3\end{bmatrix}\begin{bmatrix}0&~~~1&-1&~~~2&-2\\3&-3&~~~4&-4&~~~0\end{bmatrix} \)

20. If A=\( \begin{bmatrix}0&1&0\\0&0&1\\p&q&r\end{bmatrix} \), and I is the identity matrix of order 3 , show that

\( A^{3}=pI+qA+rA^{2} \)

21. If ​\( w \)​ is a complex cube root of unity, show that

\[ \Bigg(\begin{bmatrix}1&w&w^2\\w&w^2&1\\w^2&1&w\end{bmatrix}+\begin{bmatrix}w&w^2&1\\w^2&1&w\\w&w^2&1\end{bmatrix}\Bigg)\begin{bmatrix}1\\w\\w^2\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix} \]

22. If  A=\( \begin{bmatrix}~~~2&-3&-5\\-1&~~~4&~~~5\\~~~1&-3&-4\end{bmatrix} \), show that ​\( A^{2}=A \)

23. If A=​\( \begin{bmatrix}4&-1&-4\\3&~~~0&-4\\3&-1&-3\end{bmatrix} \)​, show that ​\( A^{2}=I_{3} \)

24. If  ​\( \begin{bmatrix}1&1&x\end{bmatrix}\begin{bmatrix}1&0&2\\0&2&1\\2&1&0\end{bmatrix}\begin{bmatrix}1\\1\\1\end{bmatrix} \)​ =0,find x.

25. If  ​\( \begin{bmatrix}x&4&1\end{bmatrix}\begin{bmatrix}2&1&2\\1&0&2\\0&2&-4\end{bmatrix}\begin{bmatrix}x\\4\\-1\end{bmatrix} \)​ =0, find x.

26. If A=

27. If A=

28. If A=

29. If A=

30. If A=

31. Show that the matrix A=

32. Show that the matrix A=

33.If A=

34.

(i) If A=

(ii) If A=

35. If A=

36. If A=

37. If A=

38. If A=

39.

40.

41. If A=

42. If A=

43. If A=

44. If A=

45. If A=

46. If A=

47.With using the concept of inverse matrix, find the matrix ​\( \begin{bmatrix}x&y\\z&u\end{bmatrix} \)​ such that

\[ \begin{bmatrix}~~~5&-7\\-2&~~~3\end{bmatrix}\begin{bmatrix}x&y\\z&u\end{bmatrix}=\begin{bmatrix}-16&-6\\~~~7&~~~2\end{bmatrix} \]

48.

49.

50. If A=

51. If A=

52. If A=

53. If A=

54. If A=

55. Let A=

56. If B, C are n rowed square matrices and if A=B+C , BC=CB , ​\( C^{2}=0 \)​ , then show that for every

57. If A=diag(a,b,c) show that ​\( A^n=diag(a^n,b^n,c^n) \)​ for all positive integers n .

58. A matrix X has a+b rows and a+2 columns while the matrix Y has b+1 rows and a+3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal.

59. Give examples of matrices

(i) A and B such that ​\( AB\not=BA \)

(ii) A and B such that AB=0 but ​\( A\not=0, B\not=0 \)

(iii) A and B such that AB=0 but ​\( BA\not=0 \)

(iv) A and B and C such that AB=AC but ​\( B\not=C,A\not=0 \)

60. Let A and B be square matrices of the same order. Does\( (A+B)^2=A^2+2AB+B^2 \)hold ? If not, why ?

61. If A and B are square matrices of the same order, explain, why in general

(i) ​\( (A+B)^2\not=A^2+2AB+B^2 \)

(ii) ​\( (A-B)^2\not=A^2-2AB+B^2 \)

(iii) ​\( (A+B)(A-B)\not=A^2-B^2 \)

62. Three shopkeeper A, B and C go to a store to buy stationary. A purchases 12 dozen notebooks,  dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebook costs 40 paise, a pen costs ₹ 1.25 and  a pencil costs 35 paise. Use matrix multiplication to calculate each individual’s bill.

63. The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are ₹ 8.30 , ₹ 3.45 , and ₹ 4.50 each respectively. Find the total amount the store will receive from selling all items.

64. In legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways : telephone , house calls and letters. The cost per contact (in paise) is given in matrix A as

A=​\( \begin{bmatrix}Cost~per ~contact\\40\\100\\50\end{bmatrix}\begin{matrix}telephone\\house~call\\letter\end{matrix} \)

The number of contacts of each type made in two cities X and Y is given in matrix B as :

B=​\( \begin{matrix}\begin{matrix}Telephone&House~call&Letter\end{matrix}\\\begin{bmatrix}1000&&&&500&&&&5000\\3000&&&&1000&&&&10000\end{bmatrix}\end{matrix}\begin{matrix}\\\rightarrow~X\\\rightarrow~Y\end{matrix} \)

Find the total amount spent by the group in the two cities X and Y.

65. A trust fund has ₹ 30,000 that must be invested in two different types of bonds.The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication determine bow to divide ₹ 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of

(i) ₹ 1800

(ii) ₹ 2000

66. Let A=​\( \begin{bmatrix}1&1&1\\3&3&3\end{bmatrix} \)​, B=​\( \begin{bmatrix}~~~3&1\\~~~5&2\\-2&4\end{bmatrix} \)​ and C=​\( \begin{bmatrix}~~~4&2\\-3&5\\~~~5&0\end{bmatrix} \)​. Verify that AB=AC though B​\( \not= \)​C , A​\( \not= \)​0

ALGEBRA OF MATRICES

EXERCISE 5.1

1.If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

solution:

Order of matrix=number of rows ​\( \times \)​number of column

or

Total number of elements=number of rows ​\( \times \)​number of column

Thus, to find all possible order of a matrix having 8 elements we have to find all possible ordered pairs whose product is 8.

Possible order pairs are

(1\( \times \)8) ; (8\( \times \)1) ; (2​\( \times \)​4) ; (4\( \times \)2) ;

AND

The possible orders of a matrix with 5 total elements are

(1 \( \times \) 5) ; (5 \( \times \) 1)

2.If 

\[ A=[​a_{ij}​]=\begin{bmatrix}2&3&-5\\1&4&~~~9\\0&7&-2\end{bmatrix}~and~B=[​b_{ij}]= \begin{bmatrix}~~~2&-1\\-3&~~~4\\~~~1&~~~2\end{bmatrix} \]

then find 

(i)​\( a_{22}+b_{21} \) ​(ii)​\( a_{11}b_{11}+a_{22}b_{22} \)

solution:

(i)\( a_{22}+b_{21} \)

where,

\( a_{22}=4 \)​and ​\( b_{21}=-3 \)

\( =a_{22}+b_{21} \)

\( =4+(-3) \)

\( =1 \)

(ii)\( a_{11}b_{11}+a_{22}b_{22} \)

where,

\( a_{11}=2~;~b_{11}=2~; \)

\( \\a_{22}=4~;~b_{22}=4 \)

\( =a_{11}b_{11}+a_{22}b_{22} \)

\( \\=(2)(2)+(4)(4) \)

\( =4+16 \)

\( =20 \)

3.Let A be a matrix of order 3​\( \times \)​4.If ​\( R_1 \)​denotes the first row of A and ​\( C_2 \)​denotes second column, then determine the orders of matrices ​\( R_1 \)​ and ​\( C_2 \)​.

4.Construct a 2​\( \times \)​3 matrix whose elements ​\( a_{ij} \)​are given by:

(i)​\( a_{ij}=i.j \)

(ii)​\( a_{ij}=2i-j \)

(iii)​\( a_{ij}=i+j \)

(iv)​\( a_{ij}=\dfrac{(i+j)^2}{2} \)

5.Construct a 2 ​\( \times \)​2 matrix whose elements ​\( a_{ij} \)​are given by:

(i)\( a_{ij}=\dfrac{(i+j)^2}{2} \)

(ii) \( a_{ij}=\dfrac{(i-j)^2}{2} \)

(iii)\( a_{ij}=\dfrac{(i-2j)^2}{2} \)​​

(iv)\( a_{ij}=\dfrac{(2i+j)^2}{2} \)

(v)\( a_{ij}=\dfrac{\lvert{2i-3j}\rvert}{2} \)

(vi)\( a_{ij}=\dfrac{\lvert{-3i+j}\rvert}{2} \)

6.Construct a 3 ​\( \times \)​ 4 matrix A=[​\( a_{ij} \)​] whose elements ​\( a_{ij} \)​are given by:

(i)\( a_{ij}=i+j \)

(ii)\( a_{ij}=i-j \)

(iii)\( a_{ij}=2i \)

(iv)\( a_{ij}=j \)

(v)\( a_{ij}=\dfrac{1}{2}~\lvert{-3i+j}\rvert \)

7.Construct a matrix 4​\( \times \)​3 matrix whose elements are 

(i)​\( a_{ij}=2i+\dfrac{i}{j} \)

(ii)​\( a_{ij}=\dfrac{i-j}{i+j} \)

(iii)​\( a_{ij}=i \)

8.Find x,y,a and b if ​

\[ \begin{bmatrix}3x+4y&2&x-2y\\a+b&2a-b&-1\end{bmatrix}=\begin{bmatrix}2&2&4\\5&-5&-1\end{bmatrix} \]

9.Find x,y,a and b if 

(i)

\[ \begin{bmatrix}2x-3y&a-b&3\\1&x+4y&3a+4b\end{bmatrix}=\begin{bmatrix}1&-2&3\\1&~~~6&29\end{bmatrix} \]

10.Find the value of a,b,c and d from the following equations:

\( \begin{bmatrix}2a+b&a-2b\\5c-d&4c+3d\end{bmatrix}=\begin{bmatrix}4&-3\\11&24\end{bmatrix} \)

11.For what values of x and y are the following matrices equal ?

\( A=\begin{bmatrix}2x+1&2y\\0&y^2-5y\end{bmatrix} \)​,

\( B=\begin{bmatrix}x+3&y^2+2\\0&-6\end{bmatrix} \)

12.Find x,y and z so that A=B,where

\( A=\begin{bmatrix}x-2&3&2z\\18z&y+2&6z\end{bmatrix} \)​,

\( B=\begin{bmatrix}y&z&6\\6y&x&2y\end{bmatrix} \)

13.If \( \begin{bmatrix}x&3x-y\\2x+z&3y-w\end{bmatrix}=\begin{bmatrix}3&2\\4&7\end{bmatrix} \)​,find x,y,z,w.

14.If ​\( \begin{bmatrix}x-y&z\\2x+y&w\end{bmatrix}=\begin{bmatrix}-1&4\\~~~0&5\end{bmatrix} \)​,find x,y,z,w.

15.Find the values of x and y if

\[ \begin{bmatrix}x+10&y^2+2y\\0&-4\end{bmatrix}=\begin{bmatrix}3x+4&3\\0&y^2-5y\end{bmatrix} \]

16.If

\[ \begin{bmatrix}x+3&z+4&2y-7\\4x+6&a-1&0\\b-3&3b&z+2c\end{bmatrix}=\begin{bmatrix}0&~~~6&3y-2\\2x&-3&2c-2\\2b+4&-21&0\end{bmatrix} \]

Obtain the value of a,b,c,x,y and z.

17.Give an example of
(i)a row matrix which is also a column matrix,
(ii)a diagonal matrix which is not scalar,
(iii)a triangular matrix

18.The sale figure of two car dealers during January 2007 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars,while dealer B sold 7 deluxe, 2 premium and 3 standard cars.Total sales over the 2 month periods of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars.In the same 2 month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars.Write 2​\( \times \)​3 matrices summarizing sales data for January and 2 month periods for each dealer.

EXERCISE 5.2

1.Compute the following sums:

(i)​\( \begin{bmatrix}3&-2\\1&~~~4\end{bmatrix}+\begin{bmatrix}-2&4\\~~~1&3\end{bmatrix} \)

(ii)\( \begin{bmatrix}~~~2&1&3\\~~~0&3&5\\-1&2&5\end{bmatrix}+\begin{bmatrix}1&-2&3\\2&~~~6&1\\0&-3&1\end{bmatrix} \)

2.Let A=​\( \begin{bmatrix}2&4\\3&2\end{bmatrix} \)​, B=​\( \begin{bmatrix}~~~1&3\\-2&5\end{bmatrix} \)​ and C=​\( \begin{bmatrix}-2&5\\~~~3&4\end{bmatrix} \)​.Find each of the following:

(i)2A-3B

(ii)B-4C

(iii)3A-C

(iv)3A-2B+3C

3.If A=​\( \begin{bmatrix}2&3\\5&7\end{bmatrix} \)​,B=​\( \begin{bmatrix}-1&0&2\\~~~3&4&1\end{bmatrix} \)​, C=​\( \begin{bmatrix}-1&2&3\\~~~2&1&0\end{bmatrix} \),find :

(i)A+B and B+C

(ii)2B+3A and 3C-4B

4.Let A=​\( \begin{bmatrix}-1&0&2\\~~~3&1&4\end{bmatrix} \)​, B=​\( \begin{bmatrix}0&-2&5\\1&-3&1\end{bmatrix} \)​ and C=​\( \begin{bmatrix}1&-5&2\\6&~~~0&-4\end{bmatrix} \)​. Compute 2A-3B+4C.

5.If A=diag(2 -5 9), B=diag(1 1 -4) and C=diag(-6 3 4) , find:

(i) A-2B

(ii) B+C-2A

(iii) 2A+3B-5C

6. Given the matrices

A=​\( \begin{bmatrix}2&~~~1&1\\3&-1&0\\0&~~~2&4\end{bmatrix} \)​, B=​\( \begin{bmatrix}9&7&-1\\3&5&~~~4\\2&1&~~~6\end{bmatrix} \)​ and C=​\( \begin{bmatrix}2&-4&3\\1&-1&0\\9&~~~4&5\end{bmatrix} \)

Verify that (A+B)+C=A+(B+C).

7.Find the matrices X and Y, if X+Y=​\( \begin{bmatrix}5&2\\0&9\end{bmatrix} \)​ and X-Y=​\( \begin{bmatrix}3&~~~6\\0&-1\end{bmatrix} \)

8.Find X if Y=​\( \begin{bmatrix}3&2\\1&4\end{bmatrix} \)​ and 2X+Y=​\( \begin{bmatrix}~~~1&0\\-3&2\end{bmatrix} \)

9.Find the matrices X and Y , if 2X-Y=​\( \begin{bmatrix}~~~6&-6&0\\-4&~~~2&1\end{bmatrix} \)​ and X+2Y=​\( \begin{bmatrix}~~~3&2&~~~5\\-2&1&-7\end{bmatrix} \)

10.If X-Y=​\( \begin{bmatrix}1&1&1\\1&1&0\\1&0&0\end{bmatrix} \)​ and X+Y=​\( \begin{bmatrix}~~~3&5&1\\-1&1&4\\~~~11&8&0\end{bmatrix} \)​ , find X and Y.

11.Find the matrix A, if ​\( \begin{bmatrix}1&2&-1\\0&4&~~~9\end{bmatrix} \)​+A=​\( \begin{bmatrix}~~~9&-1&4\\-2&~~~1&3\end{bmatrix} \)

12.If A=​\( \begin{bmatrix}9&1\\7&8\end{bmatrix} \)​ , B=​\( \begin{bmatrix}1&5\\7&12\end{bmatrix} \)​, find matrix C such that 5A+3B+2C is a null matrix.

13.If A=​\( \begin{bmatrix}~~~2&-2\\~~~4&~~~2\\-5&~~~1\end{bmatrix} \)​, B=​\( \begin{bmatrix}8&~~~0\\4&-2\\3&~~~6\end{bmatrix} \)​, find matrix X such that 2A+3X=5B.

14.If A=​\( \begin{bmatrix}1&-3&2\\2&~~~0&2\end{bmatrix} \)​ and B=​\( \begin{bmatrix}2&-1&-1\\1&~~~0&-1\end{bmatrix} \)​, find the matrix C such that A+B+C is zero matrix.

15.Find x, y satisfying the matrix equations:

(i)

\[ \begin{bmatrix}x-y&2&-2\\4&x&~~~6\end{bmatrix}+\begin{bmatrix}3&-2&~~~2\\1&~~~0&-1\end{bmatrix}=\begin{bmatrix}6&0&0\\5&2x+y&5\end{bmatrix} \]

(ii)

\[ \begin{bmatrix}x&y+2&z-3\end{bmatrix}+\begin{bmatrix}y&4&5\end{bmatrix}=\begin{bmatrix}4&9&12\end{bmatrix} \]

16.If ​

\[ 2\begin{bmatrix}3&4\\5&x\end{bmatrix}+\begin{bmatrix}1&y\\0&1\end{bmatrix}=\begin{bmatrix}7&0\\10&5\end{bmatrix} \]

​,find x and y.

17.Find the value of ​\( \lambda \)​, a non-zero scalar , if

\[ \lambda\begin{bmatrix}1&0&2\\3&4&5\end{bmatrix}+2\begin{bmatrix}~~~1&~~~2&3\\-1&-3&2\end{bmatrix}=\begin{bmatrix}4&4&10\\4&2&14\end{bmatrix} \]

18. (i) Find a matrix X such that 2A+B+X=O, where

A=​\( \begin{bmatrix}-1&2\\~~~3&4\end{bmatrix} \)​, B=​\( \begin{bmatrix}3&-2\\1&~~~5\end{bmatrix} \)

(ii) If A=​\( \begin{bmatrix}8&~~~0\\4&-2\\3&~~~6\end{bmatrix} \)​ and B=​\( \begin{bmatrix}~~~2&-2\\~~~4&~~~2\\-5&~~~1\end{bmatrix} \)​, then find the matrix X of order 3​\( \times \)​2 such that 2A+3X=5B.

19.Find x, y, z and t, if 

(i)

\[ 3\begin{bmatrix}x&y\\z&t\end{bmatrix}=\begin{bmatrix}~~~x&6\\-1&2t\end{bmatrix}+\begin{bmatrix}4&x+y\\z+t&3\end{bmatrix} \]

(ii)

\[ 2\begin{bmatrix}x&5\\7&y-3\end{bmatrix}+\begin{bmatrix}3&4\\1&2\end{bmatrix}=\begin{bmatrix}7&14\\15&14\end{bmatrix} \]

20.In a certain city there are 30 colleges. Each colleges has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix . Using scalar multiplication, find the total number of posts of each kind in all the colleges.

EXERCISE 5.3

1.Compute the indicated products:

(i) ​\( \begin{bmatrix}~~~a&b\\-b&a\end{bmatrix}\begin{bmatrix}a&-b\\b&a\end{bmatrix} \)

(ii) ​\( \begin{bmatrix}1&-2\\2&~~~3\end{bmatrix}\begin{bmatrix}~~~1&2&~~~3\\-3&2&-1\end{bmatrix} \)

(iii)\( \begin{bmatrix}2&3&4\\3&4&5\\4&5&6\end{bmatrix}\begin{bmatrix}1&-3&5\\0&~~~2&4\\3&~~~0&5\end{bmatrix} \)

2. Show that AB​\( \not= \)​BA in each of the following cases:

(i) A=​\( \begin{bmatrix}5&-1\\6&~~~7\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}2&1\\3&4\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}-1&~~~1&0\\~~~0&-1&1\\~~~2&~~~3&4\end{bmatrix} \)​ and B=​\( \begin{bmatrix}1&2&3\\0&1&0\\1&1&0\end{bmatrix} \)

(iii) A=​\( \begin{bmatrix}1&3&0\\1&1&0\\4&1&0\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}0&1&0\\1&0&0\\0&5&1\end{bmatrix} \)

3.Compare the products AB and BA whichever exists in each of the following cases:

(i) A=​\( \begin{bmatrix}1&-2\\2&~~~3\end{bmatrix} \)​ and B=​\( \begin{bmatrix}1&2&3\\2&3&1\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}~~~3&2\\-1&0\\-1&1\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}4&5&6\\0&1&2\end{bmatrix} \)

(iii) A=​\( \begin{bmatrix}1&-1&2&3\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}0\\1\\3\\2\end{bmatrix} \)

(iv)\( [a,b]\begin{bmatrix}c\\d\end{bmatrix}+\begin{bmatrix}a&b&c&d\end{bmatrix}\begin{bmatrix}a\\b\\c\\d\end{bmatrix} \)

4. Show that AB​\( \not \)​BA in each of the following cases:

(i) A=​\( \begin{bmatrix}1&~~~3&-1\\2&-1&-1\\3&~~~0&-1\end{bmatrix} \)​ and  B=​\( \begin{bmatrix}-2&3&-1\\-1&2&-1\\-6&9&-4\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}~~~10&-4&-1\\-11&~~~5&0\\~~~9&-5&1\end{bmatrix} \)​ and B=​\( \begin{bmatrix}1&2&1\\3&4&2\\1&3&2\end{bmatrix} \)

5. Evaluate the following:

(i)

\[ \Bigg(\begin{bmatrix}~~~1&~~~3\\-1&-4\end{bmatrix}+\begin{bmatrix}~~~3&-2\\-1&~~~1\end{bmatrix}\Bigg)\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix} \]

(ii)

\[ \begin{bmatrix}1&2&3\end{bmatrix}\begin{bmatrix}1&0&2\\2&0&1\\0&1&2\end{bmatrix}\begin{bmatrix}2\\4\\6\end{bmatrix} \]

(iii)

\[ \begin{bmatrix}1&-1\\0&~~~2\\2&~~~3\end{bmatrix}\Bigg(\begin{bmatrix}1&0&2\\2&0&1\end{bmatrix}-\begin{bmatrix}0&1&2\\1&0&2\end{bmatrix}\Bigg) \]

6. If A=​\( \begin{bmatrix}1&0\\0&1\end{bmatrix} \)​, B=​\( \begin{bmatrix}1&~~~0\\0&-1\end{bmatrix} \)​ and C=​\( \begin{bmatrix}0&1\\1&0\end{bmatrix} \),then show that ​\( A^2=B^2=C^2=l^2 \)

7. If A=​\( \begin{bmatrix}2&-1\\3&~~~2\end{bmatrix} \)​ and B=​\( \begin{bmatrix}~~~0&4\\-1&7\end{bmatrix} \)​,find ​\( 3A^2-2B+I \)

8. If A=​\( \begin{bmatrix}~~4&2\\-1&1\end{bmatrix} \)​,prove that (A-2I)(A-3I)=0

9. If A=​\( \begin{bmatrix}1&1\\0&1\end{bmatrix} \)​,show that ​\( A^2=\begin{bmatrix}1&2\\0&1\end{bmatrix} \)​and ​\( A^3=\begin{bmatrix}1&3\\0&1\end{bmatrix} \)

10. If A=​\( \begin{bmatrix}~~~ab&~~~b^2\\-a^2&-ab\end{bmatrix} \)​,show that ​\( A^2=0 \)

11. If A=​\( \begin{bmatrix}~~~cos2\theta&sin2\theta\\-sin2\theta&cos2\theta\end{bmatrix} \)​, find ​\( A^2 \)

12. If A=​\( \begin{bmatrix}~~~2&-3&-5\\-1&~~~4&~~~5\\~~~1&-3&-4\end{bmatrix} \)​ and B=​\( \begin{bmatrix}-1&~~~3&~~~5\\~~~1&-3&-5\\-1&~~~3&~~~5\end{bmatrix} \)​show that Ab=BA=​\( O_{{3}\times{3}} \)

13. If A=​\( \begin{bmatrix}~~~0&~~~c&-b\\-c&~~~0&~~~a\\~~~b&-a&~~~0\end{bmatrix} \)​ and B=​\( \begin{bmatrix}a^2&ab&ac\\ab&b^2&bc\\ac&bc&c^2\end{bmatrix} \)​show that AB=BA=​\( O_{{3}\times{3}} \)

14. If A=​\( \begin{bmatrix}2&-3&-5\\-1&~~~4&~~~5\\1&-3&-4\end{bmatrix} \)​ and B=​\( \begin{bmatrix}~~~2&-2&~~~4\\-1&~~~3&~~~4\\~~~1&-2&-3\end{bmatrix} \)​, show that AB=A and BA=B.

15. Let A=​\( \begin{bmatrix}-1&~~~1&-1\\~~~3&-3&~~~3\\~~~5&~~~5&~~~5\end{bmatrix} \)​ and B=​\( \begin{bmatrix}~~~0&~~~4&~~~3\\~~~1&-3&-3\\-1&~~~4&~~~4\end{bmatrix} \)​, compute ​\( A^2-B^2 \)

16. For the following matrices verify the associativity of matrix multiplicationi.e. (AB)C=A(BC):

(i) A=\( \begin{bmatrix}~~~1&~~2&~~0\\-1&~~0&~~~1\end{bmatrix} \), B=​\( \begin{bmatrix}~~~1&0\\-1&2\\~~~0&3\end{bmatrix} \)​ and C=​\( \begin{bmatrix}~~~1\\-1\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}4&2&3\\1&1&2\\3&0&2\end{bmatrix} \)​, B=​\( \begin{bmatrix}1&-1&1\\0&~~~1&2\\2&-1&1\end{bmatrix} \)​ , C=​\( \begin{bmatrix}1&2&-1\\3&0&~~~1\\0&0&~~~1\end{bmatrix} \)

17. For the following matrices verify the distributivity of the matrix multiplication over matrix addition i.e. A(B+C)=AB+AC

(i) A=​\( \begin{bmatrix}1&-1\\0&~~~2\end{bmatrix} \)​ , B=​\( \begin{bmatrix}-1&0\\~~~2&1\end{bmatrix} \)​ , C=​\( \begin{bmatrix}0&~~~1\\1&-1\end{bmatrix} \)

(ii) A=​\( \begin{bmatrix}~~~2&-1\\~~~1&~~~1\\-1&~~~2\end{bmatrix} \)​ , B=​\( \begin{bmatrix}0&1\\1&1\end{bmatrix} \)​ , C=​\( \begin{bmatrix}1&-1\\0&~~~1\end{bmatrix} \)

18. If A=​\( \begin{bmatrix}~~~1&~~~0&-2\\~~~3&-1&~~~0\\-2&~~~1&~~~1\end{bmatrix} \)​ , B=​\( \begin{bmatrix}~~~0&5&-4\\-2&1&~~~3\\-1&0&~~~2\end{bmatrix} \)​ , C=​\( \begin{bmatrix}~~~1&5&2\\-1&1&0\\~~~0&-1&1\end{bmatrix} \)

19.Compute the following elements ​\( a_{43} \)​ and ​\( a_{22} \)​ of the matrix :

A=​\( \begin{bmatrix}0&1&0\\2&0&2\\0&3&2\\4&0&4\end{bmatrix}\begin{bmatrix}~~~2&-1\\-3&~~~2\\~~~4&~~~3\end{bmatrix}\begin{bmatrix}0&~~~1&-1&~~~2&-2\\3&-3&~~~4&-4&~~~0\end{bmatrix} \)

20. If A=\( \begin{bmatrix}0&1&0\\0&0&1\\p&q&r\end{bmatrix} \), and I is the identity matrix of order 3 , show that

\( A^{3}=pI+qA+rA^{2} \)

21. If ​\( w \)​ is a complex cube root of unity, show that

\[ \Bigg(\begin{bmatrix}1&w&w^2\\w&w^2&1\\w^2&1&w\end{bmatrix}+\begin{bmatrix}w&w^2&1\\w^2&1&w\\w&w^2&1\end{bmatrix}\Bigg)\begin{bmatrix}1\\w\\w^2\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix} \]

22. If  A=\( \begin{bmatrix}~~~2&-3&-5\\-1&~~~4&~~~5\\~~~1&-3&-4\end{bmatrix} \), show that ​\( A^{2}=A \)

23. If A=​\( \begin{bmatrix}4&-1&-4\\3&~~~0&-4\\3&-1&-3\end{bmatrix} \)​, show that ​\( A^{2}=I_{3} \)

24. If  ​\( \begin{bmatrix}1&1&x\end{bmatrix}\begin{bmatrix}1&0&2\\0&2&1\\2&1&0\end{bmatrix}\begin{bmatrix}1\\1\\1\end{bmatrix} \)​ =0,find x.

25. If  ​\( \begin{bmatrix}x&4&1\end{bmatrix}\begin{bmatrix}2&1&2\\1&0&2\\0&2&-4\end{bmatrix}\begin{bmatrix}x\\4\\-1\end{bmatrix} \)​ =0, find x.

26. If A=

27. If A=

28. If A=

29. If A=

30. If A=

31. Show that the matrix A=

32. Show that the matrix A=

33.If A=

34.

(i) If A=

(ii) If A=

35. If A=

36. If A=

37. If A=

38. If A=

39.

40.

41. If A=

42. If A=

43. If A=

44. If A=

45. If A=

46. If A=

47.With using the concept of inverse matrix, find the matrix ​\( \begin{bmatrix}x&y\\z&u\end{bmatrix} \)​ such that

\[ \begin{bmatrix}~~~5&-7\\-2&~~~3\end{bmatrix}\begin{bmatrix}x&y\\z&u\end{bmatrix}=\begin{bmatrix}-16&-6\\~~~7&~~~2\end{bmatrix} \]

48.

49.

50. If A=

51. If A=

52. If A=

53. If A=

54. If A=

55. Let A=

56. If B, C are n rowed square matrices and if A=B+C , BC=CB , ​\( C^{2}=0 \)​ , then show that for every

57. If A=diag(a,b,c) show that ​\( A^n=diag(a^n,b^n,c^n) \)​ for all positive integers n .

58. A matrix X has a+b rows and a+2 columns while the matrix Y has b+1 rows and a+3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal.

59. Give examples of matrices

(i) A and B such that ​\( AB\not=BA \)

(ii) A and B such that AB=0 but ​\( A\not=0, B\not=0 \)

(iii) A and B such that AB=0 but ​\( BA\not=0 \)

(iv) A and B and C such that AB=AC but ​\( B\not=C,A\not=0 \)

60. Let A and B be square matrices of the same order. Does\( (A+B)^2=A^2+2AB+B^2 \)hold ? If not, why ?

61. If A and B are square matrices of the same order, explain, why in general

(i) ​\( (A+B)^2\not=A^2+2AB+B^2 \)

(ii) ​\( (A-B)^2\not=A^2-2AB+B^2 \)

(iii) ​\( (A+B)(A-B)\not=A^2-B^2 \)

62. Three shopkeeper A, B and C go to a store to buy stationary. A purchases 12 dozen notebooks,  dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebook costs 40 paise, a pen costs ₹ 1.25 and  a pencil costs 35 paise. Use matrix multiplication to calculate each individual’s bill.

63. The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are ₹ 8.30 , ₹ 3.45 , and ₹ 4.50 each respectively. Find the total amount the store will receive from selling all items.

64. In legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways : telephone , house calls and letters. The cost per contact (in paise) is given in matrix A as

A=​\( \begin{bmatrix}Cost~per ~contact\\40\\100\\50\end{bmatrix}\begin{matrix}telephone\\house~call\\letter\end{matrix} \)

The number of contacts of each type made in two cities X and Y is given in matrix B as :

B=​\( \begin{matrix}\begin{matrix}Telephone&House~call&Letter\end{matrix}\\\begin{bmatrix}1000&&&&500&&&&5000\\3000&&&&1000&&&&10000\end{bmatrix}\end{matrix}\begin{matrix}\\\rightarrow~X\\\rightarrow~Y\end{matrix} \)

Find the total amount spent by the group in the two cities X and Y.

65. A trust fund has ₹ 30,000 that must be invested in two different types of bonds.The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication determine bow to divide ₹ 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of

(i) ₹ 1800

(ii) ₹ 2000

66. Let A=​\( \begin{bmatrix}1&1&1\\3&3&3\end{bmatrix} \)​, B=​\( \begin{bmatrix}~~~3&1\\~~~5&2\\-2&4\end{bmatrix} \)​ and C=​\( \begin{bmatrix}~~~4&2\\-3&5\\~~~5&0\end{bmatrix} \)​. Verify that AB=AC though B​\( \not= \)​C , A​\( \not= \)​0

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