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

Question

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

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

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

B = [(1,1,2)]

C = [1 - 2]

Answer 1

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

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

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 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 

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

thus matrix BC is of order 3 x 2

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

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

Matrix A(BC) = \(\begin{bmatrix} 3 & -6 \\[0.3em] 7 & -14 \\[0.3em] \end{bmatrix}\)

Matrix A(BC) = \(\begin{bmatrix} 3 & -6 \\[0.3em] 7 & -14 \\[0.3em] \end{bmatrix}\)

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

thus matrix BC is of order 2 x 1

Matrix AB = \(\begin{bmatrix} 3 \\[0.3em] 7 \\[0.3em] \end{bmatrix}\)

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

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

Matrix A(BC) = (AB)C = \(\begin{bmatrix} 3&-6 \\[0.3em] 7 & -14 \\[0.3em] \end{bmatrix}\)