Verify that A(B + C) = (AB + AC), when A = [(1,2)(3,4)], B = [(2,0)(1,-3)] and C = [(1,-1)(0,1)].

Question

Verify that A(B + C) = (AB + AC), when

A = \(\begin{bmatrix} 1 & 2 \\[0.3em] 3& 4 \\[0.3em] \end{bmatrix}\), B = \(\begin{bmatrix} 2& 0 \\[0.3em] 1& -3 \\[0.3em] \end{bmatrix}\) and C = \(\begin{bmatrix} 1&-1 \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix}.\) 

A = [(1,2)(3,4)],

B = [(2,0)(1,-3)]

C = [(1,-1)(0,1)].

Answer 1

Given : A = \(\begin{bmatrix} 1 & 2 \\[0.3em] 3& 4 \\[0.3em] \end{bmatrix}\), B = \(\begin{bmatrix} 2& 0 \\[0.3em] 1& -3 \\[0.3em] \end{bmatrix}\) and C = \(\begin{bmatrix} 1&-1 \\[0.3em] 0& 1 \\[0.3em] \end{bmatrix}.\)

Matrix A is of order 2 x 2 , matrix B is of order 2 x 2 and matrix C is of order 2 x 2

To verify : A(B + C) = (AB + AC)

Formula used :

Where c_ij = a_i1b_1j + a_i2b_2j + a_i3b_3j + ……………… + a_inb_nj

If A is a matrix of order a b and B is a matrix of order c x d ,then matrix AB exists and is of order a x d ,

if and only if b = c

If A is a matrix of order a b and B is a matrix of order c x d ,then matrix BA exists and is of order c x b ,

if and only if d = a

Matrix A(B + C) is of order 2 x 2

For matrix AB, a = b = c = d = 2 ,matrix AB is of order 2 x 2

A(B + C) = \(\begin{bmatrix} 5 & -5 \\[0.3em] 13 & -11 \\[0.3em] \end{bmatrix}\)

For matrix AB, a = b = c = d = 2 ,matrix AB is of order 2 x 2

For matrix AC, a = b = c = d = 2 ,matrix AC is of order 2 x 2

A(B + C) = (AB + AC)