Question

If A = $\left[\begin{array}{cc}1& 2\\ 3& 0\\ 4& 1\end{array}\right]$ and B = $\left[\begin{array}{ccc}0& 1& 0\\ 0& 2& 1\\ 2& 3& 1\end{array}\right]$ find BA. Can we find AB also?

Answer 1

A = 3 $\times$ 2 and B = 3 $\times$ 3. Since the number of columns of A is not equal to number of rows of B as such the matrix AB is not defined. But the matrix BA is defined because the number of columns in B is equal to the number of rows in A each being equal to 3.
BA = $\left[\begin{array}{ccc}0& 1& 0\\ 0& 2& 1\\ 2& 3& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 0\\ 4& 1\end{array}\right]$
= $\left[\begin{array}{cc}0.1+1.3+0.4& 0.2+1.0+0.1\\ 0.1+2.3+1.4& 0.2+2.0+1.1\\ 2.1+3.3+1.4& 2.2+3.0+1.1\end{array}\right]=\left[\begin{array}{cc}3& 0\\ 10& 1\\ 15& 5\end{array}\right]$
Verification for the Product to be correct.
We have proved above that
BA = $\left[\begin{array}{ccc}0& 1& 0\\ 0& 2& 1\\ 2& 3& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 0\\ 4& 1\end{array}\right]=\left[\begin{array}{cc}3& 0\\ 10& 1\\ 15& 5\end{array}\right]$
sum 2 6 2 28 6
Multiply the row of sum i.e. 2,6,2 with the columns of 2nd matrix and the result should come out to be row of sum of product matrix on the right i.e. 28,6 as shown below.
i.e. 2.1 + 6.3 + 2.4 = 2 + 18 + 8 = 28
and 2.2 + 6.0 + 2.1 = 4 + 0 + 2 = 6
Hence correct.