Rd sharma solution Chapter 5 Algebra of matrices

Exercise 5.1 Exercise 5.2  Exercise 5.3

EXERCISE 5.1

Question 1

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

Sol:

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)

 


 

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}

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

 


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}

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}

 


 

Question 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


 

Question 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

 


 

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:

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

Sol :

 


 

Question 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}

Sol :

 

 


 

Question 12

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

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

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

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

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.

Sol:

 


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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