OPERATIONS WITH MATRICES

Equality of Matrices :

Two matrices A  =  [a_ij]_mxn and B  =  [bij]_mxn are said to be equal if 

(i)  They are of same order and

(ii) Each element of A is equal to the corresponding element of B, that is a_ij  =  b_ij for all i and j.

For example, the matrices

are not equal as the orders of the matrices are different.

Multiplication of a matrix by a scalar :

For a given matrix A  =  [a_ij]_mxn and a scalar (real number) k, we define a new matrix B  =  [b_ij]_mxn, where

b_ij  =  ka_ij

for all i and j.

Thus, the matrix B is obtained by multiplying each entry of A by the scalar k and written as B = kA. This multiplication is called scalar multiplication.

(iii) Addition of matrices :

If A  =  [a_ij]_mxn and B  =  [b_ij]_mxn are two matrices of the same order, then the addition of A and B is a matrix C  =  [c_ij]_mxn, where

c_ij  =  a_ij + b_ij 

for all i and j.

The addition of two matrices A and B is denoted by A+B. Addition is not defined for matrices of different orders.

(iv) Negative of a matrix :

The negative of a matrix A  =  [a_ij]_mxn is denoted by -A and is defined as -A = (- 1)A.

That is,

-A  =  [b_ij]mxn, where b_ij = a_ij

for all i and j.

(v) Subtraction of matrices :

If A  =  [a_ij]_mxn B  =  [b_ij]_mxn are two matrices of the same order, then subtraction A - B is defined as A - B  =  [c_ij]_mxn, where

c_ij  =  a_ij - b_ij 

for all i and j.

Operations with Matrices - Examples

Question 1 :

Find the values of x, y and z from the matrix equation

Solution :

The above matrices on both sides same order. So, the corresponding elements are equal.

5x + 2  =  12  -----(1)

y - 4  =  -8  -----(2)

4z + 6  =  2  -----(3)

By solving (1), we get

So, the values of x, y and z are 2, -4 and -1 respectively.

Question 2 :

Let 

Find the matrix C if C = 2A + B

Solution :

To get the new matrix C, we should multiply the scalar 2 by matrix A and add it by the matrix B.

Question 3 :

Solution :

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