Question

Compute the products AB and BA whichever exists in the following cases:
\[A=\left[ \begin{matrix}
   1 & -2 \\
   2 & 3 \\
\end{matrix} \right],B=\left[ \begin{align}
  & \begin{matrix}
   1 & 2 & 3 \\
\end{matrix} \\
 & \begin{matrix}
   2 & 3 & 1 \\
\end{matrix} \\
\end{align} \right]\]

Answer 1

Hint: The product of the two matrices with $m\times n$ and $a\times b$ number of rows and columns respectively exist when the number of columns of the first matrix is equal to the number of rows of the second matrix. This means $n=a$. If this condition is following then proceed in multiplying the two matrices.

Complete step by step solution:
The two matrices given in the above problem is as follows:
\[A=\left[ \begin{matrix}
   1 & -2 \\
   2 & 3 \\
\end{matrix} \right],B=\left[ \begin{align}
  & \begin{matrix}
   1 & 2 & 3 \\
\end{matrix} \\
 & \begin{matrix}
   2 & 3 & 1 \\
\end{matrix} \\
\end{align} \right]\]
First matrix contains 2 rows and 2 columns so this matrix is $2\times 2$ and the second matrix B contains 2 rows and 3 columns so the second matrix B is of the form $2\times 3$.
Now, if we multiply A and B then we have to multiply the matrix $2\times 2$ by the matrix $2\times 3$. As you can see, the number of columns of the first row is equal to the number of rows of the second matrix so we can compute the product of A and B.
\[\begin{align}
  & AB=\left[ \begin{matrix}
   1 & -2 \\
   2 & 3 \\
\end{matrix} \right]\left[ \begin{matrix}
   \begin{matrix}
   1 & 2 & 3 \\
\end{matrix} \\
   \begin{matrix}
   2 & 3 & 1 \\
\end{matrix} \\
\end{matrix} \right] \\
 & \Rightarrow AB=\left[ \begin{matrix}
   \begin{matrix}
   1\left( 1 \right)-2\left( 2 \right) & 1\left( 2 \right)-2\left( 3 \right) & 1\left( 3 \right)-2\left( 1 \right) \\
\end{matrix} \\
   \begin{matrix}
   2\left( 1 \right)+3\left( 2 \right) & 2\left( 2 \right)+3\left( 3 \right) & 2\left( 3 \right)+3\left( 1 \right) \\
\end{matrix} \\
\end{matrix} \right] \\
 & \Rightarrow AB=\left[ \begin{matrix}
   \begin{matrix}
   1-4 & 2-6 & 3-2 \\
\end{matrix} \\
   \begin{matrix}
   2+6 & 4+9 & 6+3 \\
\end{matrix} \\
\end{matrix} \right] \\
 & \Rightarrow AB=\left[ \begin{matrix}
   \begin{matrix}
   -3 & -4 & 1 \\
\end{matrix} \\
   \begin{matrix}
   8 & 13 & 9 \\
\end{matrix} \\
\end{matrix} \right] \\
\end{align}\]
Hence, we have computed the product AB.
Now, can we compute the product BA. The matrix B is of the form $2\times 3$ and matrix A is of the form $2\times 2$ and as you can see that the number of columns of matrix B is 3 and number of rows of the matrix A is 2 and these two are not equal so computation of the product BA is not possible.

Note: So, whenever we asked to compute the product of the two matrices, first of all check whether the multiplication is possible or not by seeing the number of columns of the first matrix and the number of rows of the second matrix if they are equal then multiplication is possible otherwise not.
Also, from this problem, we have learnt that in an algebra product of $ab$ or $ba$ is the same but in the matrix multiplication this is not the case AB might be equal or not equal to BA.