Question

If $A=\left[\begin{array}{cc}1& 2\\ 3& -4\\ 5& 6\end{array}\right]$ and $B=\left[\begin{array}{ccc}4& 5& 6\\ 7& -8& 2\end{array}\right]$ will $AB$ be equal to $BA$. Also find $AB$ & $BA$.

Answer 1

We are given,

$A={\left[\begin{array}{cc}1& 2\\ 3& -4\\ 5& 6\end{array}\right]}_{3\times 2}$ and $B={\left[\begin{array}{ccc}4& 5& 6\\ 7& -8& 2\end{array}\right]}_{2\times 3}$

here the product $A$ & $B$ gives us $3\times 3$ Matrix and $B$ & $A$ gives as $2\times 2$ Matrix.

So, $AB\ne BA$.

Now, $AB={\left[\begin{array}{cc}1& 2\\ 3& -4\\ 5& 6\end{array}\right]}_{3\times 2}{\left[\begin{array}{ccc}4& 5& 6\\ 7& -8& 2\end{array}\right]}_{2\times 3}$

$AB={\left[\begin{array}{ccc}4+14& 5-16& 6+4\\ 12-28& 15+32& 18-8\\ 20+42& 25-48& 30+12\end{array}\right]}_{3\times 3}$

$AB={\left[\begin{array}{ccc}16& -11& 10\\ -16& 47& 10\\ 62& -23& 42\end{array}\right]}_{3\times 3}$ $⟶\left(1\right)$

$BA={\left[\begin{array}{ccc}4& 5& 6\\ 7& -8& 2\end{array}\right]}_{2\times 3}{\left[\begin{array}{cc}1& 2\\ 3& -4\\ 5& 6\end{array}\right]}_{3\times 2}$

$={\left[\begin{array}{cc}4+15+30& 8-20+36\\ 7-24+10& 14+32+11\end{array}\right]}_{2\times 2}$

${\left[\begin{array}{cc}49& 24\\ -7& 58\end{array}\right]}_{2\times 2}$ $⟶\left(2\right)$

So, $ABneqBA$.

From $\left(1\right)$ & $\left(2\right)$