91Ó°ÊÓ

Matrix multiplication is abinary operation that creates a matrix by multiplying two matrices together. The number of columns in the first matrix must equal the number of rows in the second matrix for matrix multiplication to work.

For example, if matrix and is defined as $\left(\text{3}×\text{2}\right)$and $\left(\text{3}×\text{2}\right)$then, the product of the matrices and is meaningless as the columns in the first matrix (here, 2) is not equals to the number of rows in the second matrix (here, 3).

The given matrices are $A=\left(\begin{array}{cc}1& 2\\ 3& 6\end{array}\right),B=\left(\begin{array}{cc}10& 4\\ 5& -2\end{array}\right)$.

The products of two of meaningful matrices is to be find.

Find the product of the matrix AB.

$\begin{array}{rcl}AB& =& \left(\begin{array}{cc}1& 2\\ 3& 6\end{array}\right)\left(\begin{array}{cc}10& 4\\ 5& -2\end{array}\right)\\ & =& \left(\begin{array}{cccc}10& -10& 4& -4\\ 30& -30& 12& -12\end{array}\right)\\ & =& \left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\end{array}$

Find the product of the matrix BA.

$\begin{array}{rcl}BA& =& \left(\begin{array}{cc}10& 4\\ -5& -2\end{array}\right)\left(\begin{array}{cc}1& 2\\ 3& 6\end{array}\right)\\ & =& \left(\begin{array}{cc}22& 44\\ -11& -22\end{array}\right)\end{array}$

To show thatis singular,$\text{det}\left(\text{A}\right)=\text{0}$.

$\begin{array}{rcl}\text{det}\left(\text{A}\right)& =& \left|\begin{array}{cc}1& 2\\ 3& 6\end{array}\right|\\ & =& 6-2×3\\ & =& 0\end{array}$

Therefore, matrix is singular.