91Ó°ÊÓ

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

For example, if matrix A and B is defined as (3X2) and (3X2) then, the product of the matrices A and B is meaningless as the columns in the first matrix (here, 2) is not equal to the number of rows in the second matrix (here, 3).

The given matrices are

$\mathrm{A}=\left(23\right),\mathrm{B}=\left(\begin{array}{cc}-1& 4\\ 2& -1\end{array}\right),\mathrm{C}=\left(\begin{array}{c}-1\\ 2\end{array}\right)$

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

Find the product of the matrix AB.

$$\begin{gathered} \mathrm{AB}=\left(23\right)\left(\begin{array}{cc}-1& 4\\ 2& -1\end{array}\right) \\ =\left(\left(2×\right)\left(-1\right)\right)+\left(3×2\right)\left(2×4\right)+\left(3×\left(-1\right)\right)) \\ =\left(45\right) \end{gathered}$$

Find the product of the matrix (AB)C .

$$\begin{gathered} \left(\mathrm{AB}\right)\mathrm{C}=\left(45\right)\left(\begin{array}{c}-1\\ 2\end{array}\right) \\ =6 \end{gathered}$$

Find the product of the matrix BC.

$$\begin{gathered} \mathrm{BC}=\left(\begin{array}{cc}-1& 4\\ 2& -1\end{array}\right)\left(\begin{array}{c}-1\\ 2\end{array}\right) \\ =\left(\begin{array}{c}9\\ -4\end{array}\right) \end{gathered}$$

Find the product of the matrix A(BC).

$$\begin{gathered} \mathrm{A}\left(\mathrm{BC}\right)=\left(23\right)\left(\begin{array}{c}9\\ -4\end{array}\right) \\ =6 \end{gathered}$$

Therefore, matrix multiplication is associative, that is A(BC)=(AB)C_.