For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC). A = [(4,2,3)(1,1,2)(3,0,1)]

Question

For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC).

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

Answer 1

Given,

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

A(BC) = \(\begin{bmatrix} -5 & 20 & -4 \\[0.3em] -1 & 10 &-2 \\[0.3em] -7 & 10 & -5 \end{bmatrix}\) ...(2)

From equation (1) and (2),

(AB)C = A(BC)