Question

If $A=\left[\begin{array}{ccc}1& 4& 0\\ 2& 5& 0\\ 3& 6& 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}3& 2& 1\\ 1& 2& 3\\ 4& 5& 6\end{array}\right]$ and $C=\left[\begin{array}{ccc}3& 2& 1\\ 1& 2& 3\\ 7& 8& 9\end{array}\right]$, Then evaluate $AB-BC$

Answer 1

$A=\left[\begin{array}{ccc}1& 4& 0\\ 2& 5& 0\\ 3& 6& 0\end{array}\right]$

$\Rightarrow B=\left[\begin{array}{ccc}3& 2& 1\\ 1& 2& 3\\ 2& 5& 6\end{array}\right]$

$\Rightarrow C=\left[\begin{array}{ccc}3& 2& 1\\ 1& 2& 3\\ 7& 8& 9\end{array}\right]$

$\Rightarrow AB=\left[\begin{array}{ccc}1& 4& 0\\ 2& 5& 0\\ 3& 6& 0\end{array}\right]\left[\begin{array}{ccc}3& 2& 1\\ 1& 2& 3\\ 2& 5& 6\end{array}\right]$

$=\left[\begin{array}{ccc}3+4+0& 2+8+0& 1+12+0\\ 6+5+0& 4+10+0& 2+12+0\\ 9+6+0& 6+12+0& 3+18+0\end{array}\right]$

$=\left[\begin{array}{ccc}7& 10& 13\\ 11& 14& 17\\ 15& 18& 21\end{array}\right]$

$\Rightarrow BC=\left[\begin{array}{ccc}3& 2& 1\\ 1& 2& 3\\ 2& 5& 6\end{array}\right]\left[\begin{array}{ccc}3& 2& 1\\ 1& 2& 3\\ 7& 8& 9\end{array}\right]$

$=\left[\begin{array}{ccc}9+2+7& 6+4+8& 3+6+9\\ 3+2+21& 2+4+24& 1+6+27\\ 6+5+42& 4+10+48& 2+15+54\end{array}\right]$

$\Rightarrow BC=\left[\begin{array}{ccc}18& 18& 18\\ 26& 30& 34\\ 53& 62& 71\end{array}\right]$

$\Rightarrow AB-BC=\left[\begin{array}{ccc}7& 10& 13\\ 11& 14& 17\\ 15& 18& 21\end{array}\right]-\left[\begin{array}{ccc}18& 18& 18\\ 26& 30& 34\\ 53& 62& 71\end{array}\right]$

$=\left[\begin{array}{ccc}-11& -8& -5\\ -15& -16& -17\\ -38& -44& -50\end{array}\right]$

$=(-1)\left[\begin{array}{ccc}11& 8& 5\\ 15& 16& 17\\ 38& 44& 50\end{array}\right]$

Hence, the answer is $(-1)\left[\begin{array}{ccc}11& 8& 5\\ 15& 16& 17\\ 38& 44& 50\end{array}\right].$