Question

Verify that \( ( AB ) C = A ( BC ) \) if \( A=\left[\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] \) and \( C=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 0\end{array}\right] \)

Answer 1

A = \(\begin{bmatrix}1&2\\3&1\end{bmatrix}\) B = \(\begin{bmatrix}2&1\\3&2\end{bmatrix},\) C = \(\begin{bmatrix}1&2&3\\3&1&0\end{bmatrix}\) 

AB = \(\begin{bmatrix}8&5\\9&5\end{bmatrix}\)

(AB)C = \(\begin{bmatrix}23&21&24\\24&23&27\end{bmatrix}\) 

BC = \(\begin{bmatrix}5&5&6\\9&8&9\end{bmatrix}\)

A(BC) = \(\begin{bmatrix}23&21&24\\24&23&27\end{bmatrix}\)

Hence (AB) C = A(BC)

Hence verified.