Question

If A = $\left[\begin{array}{ccc}2&1\\1&3\end{array}\right]$ and B = $\left[\begin{array}{ccc}3&2&0\\1&0&4\end{array}\right]$ , find AB. Find BA, if exists?

Answer 1

Answer:

$AB=\left[\begin{array}{ccc}7&4&4\\6&2&12\end{array}\right]$

B.A does not exist

Step-by-step explanation:

If

$A=\left[\begin{array}{ccc}2&1\\1&3\end{array}\right]_{2\times2}\\\\\\B=\left[\begin{array}{ccc}3&2&0\\1&0&4\end{array}\right]_{2\times3}$

As we know that matrix multiplication is possible only if number of columns of first matrix is equal to the number of rows of second matrix.

Thus AB is possible.

$AB=\left[\begin{array}{ccc}2&1\\1&3\end{array}\right]\times \left[\begin{array}{ccc}3&2&0\\1&0&4\end{array}\right] \\\\\\= \left[\begin{array}{ccc}2(3)+1(1)&2(2)+1(0)&2(0)+1(4)\\1(3)+3(1)&1(2)+(0)(3)&1(0)+3(4)\end{array}\right] \\\\\\AB= \left[\begin{array}{ccc}7&4&4\\6&2&12\end{array}\right]$

As on calculating BA,here number of columns in B are 3 and number of rows in matrix A is 2,thus BA does not exist.and we know that matrix multiplication is not commutative.