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Rd sharma solution Chapter 5 Algebra of matrices

Exercise 5.1 Exercise 5.2  Exercise 5.3


Question 1

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


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


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) ;


The possible orders of a matrix with 5 total elements are

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



Question 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}

Sol :

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


a_{22}=4 and b_{21}=-3


= 4 + (-3)

= 1


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





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

= 4 + 16

= 20


Question 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



Question 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}


Question 5

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

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














Question 6

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







Question 7

Construct a matrix 4 \times 3 matrix whose elements are 

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






Question 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}

Sol :



Question 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}



Question 10

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


Sol :



Question 11

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



Sol :




Question 12

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



Sol :




Question 13

If  \begin{bmatrix}x&3x-y\\2x+z&3y-w\end{bmatrix}=\begin{bmatrix}3&2\\4&7\end{bmatrix}

Sol :



Question 14

If \begin{bmatrix}x-y&z\\2x+y&w\end{bmatrix}=\begin{bmatrix}-1&4\\~~~0&5\end{bmatrix}

Sol :




Question 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}

Sol :



Question 16



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



Question 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


Question 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.

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