For the following matrices, verify that A(BC) = (AB)C : A = [(1,2,5)(0,1,3)], B = [(2,3,0)(1,0,4)(1,-1,2)] and C = [(1,4,5)]

Question

For the following matrices, verify that A(BC) = (AB)C :

A = \(\begin{bmatrix} 1& 2& 5 \\[0.3em] 0& 1 &3\\[0.3em] \end{bmatrix}\), B = \(\begin{bmatrix} 2& 3 & 0 \\[0.3em] 1& 0 & 4\\[0.3em] 1 & -1& 2 \end{bmatrix}\) and C = \(\begin{bmatrix} 1 \\[0.3em] 4 \\[0.3em] 5 \end{bmatrix}\)

A = [(1,2,5)(0,1,3)],

B = [(2,3,0)(1,0,4)(1,-1,2)]

C = [(1,4,5)]

Answer 1

Given : A = \(\begin{bmatrix} 1& 2& 5 \\[0.3em] 0& 1 &3\\[0.3em] \end{bmatrix}\), B = \(\begin{bmatrix} 2& 3 & 0 \\[0.3em] 1& 0 & 4\\[0.3em] 1 & -1& 2 \end{bmatrix}\) and C = \(\begin{bmatrix} 2 \\[0.3em] 4 \\[0.3em] 5 \end{bmatrix}\)

Matrix A is of order 2 x 3 , matrix B is of order 3 x 3 and matrix C is of order 3 x 1

To show : matrix A(BC) = (AB)C

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 x 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

For matrix BC, a = 3,b = c = 3,d = 1 ,

thus matrix BC is of order 3 x 1

Matrix BC = \(\begin{bmatrix} 14 \\[0.3em] 21 \\[0.3em] 7 \end{bmatrix}\)

For matrix A(BC),a = 2 ,b = c = 3 ,d = 1 ,

thus matrix A(BC) is of order 2 x 1

Matrix A(BC) = \(\begin{bmatrix} 91 \\[0.3em] 42 \\[0.3em] \end{bmatrix}\)

Matrix A(BC) = \(\begin{bmatrix} 91 \\[0.3em] 42 \\[0.3em] \end{bmatrix}\)

For matrix AB, a = 2,b = c = 3,d = 3 ,

thus matrix BC is of order 2 x 3

For matrix (AB)C, a = 2,b = c = 3,d = 1 ,

thus matrix (AB)C is of order 2 x 1

Matrix (AB)C = \(\begin{bmatrix} 91 \\[0.3em] 42 \\[0.3em] \end{bmatrix}\)

Matrix A(BC) = (AB)C = \(\begin{bmatrix} 91 \\[0.3em] 42 \\[0.3em] \end{bmatrix}\)