Question

Find each product, if possible.

$\left[\begin{array}{cc}0& 8\\ 3& 1\\ -1& 5\end{array}\right]\cdot \left[\begin{array}{ccc}3& 1& -2\\ 0& 8& -5\end{array}\right]$

Short answer

The required matrix is,

$\left[\begin{array}{ccc}0& 64& -40\\ 9& 11& -11\\ -3& 39& -23\end{array}\right]$

Step-by-step solution

Step 1- Make the first row of product matrix

Write the multiplication of first row of first matrix and each column of second matrix and perform addition between the multiplications to form the first row of product matrix.

$\left[\begin{array}{ccc}0\cdot 3+8\cdot 0& 0\cdot 1+8\cdot 8& 0\cdot (-2)+8\cdot (-5)\\ & & \\ & & \end{array}\right]$

– Make the second row of product matrix

Write the multiplication of second row of first matrix and each column of second matrix and perform addition between the multiplications to form the second row of product matrix

$\left[\begin{array}{ccc}0\cdot 3+8\cdot 0& 0\cdot 1+8\cdot 8& 0\cdot (-2)+8\cdot (-5)\\ 3\cdot 3+1\cdot 0& 3\cdot 1+1\cdot 8& 3\cdot (-2)+1\cdot (-5)\\ & & \end{array}\right]$

– Make the third row of product matrix

Write the multiplication of third row of first matrix and each column of second matrix and perform addition between the multiplications to form the third row of product matrix

$\left[\begin{array}{ccc}0\cdot 3+8\cdot 0& 0\cdot 1+8\cdot 8& 0\cdot (-2)+8\cdot (-5)\\ 3\cdot 3+1\cdot 0& 3\cdot 1+1\cdot 8& 3\cdot (-2)+1\cdot (-5)\\ (-1)\cdot 3+5\cdot 0& (-1)\cdot 1+5\cdot 8& (-1)\cdot (-2)+5\cdot (-5)\end{array}\right]$

Step 3- Write the actual product matrix

Perform simplification of obtained matrix.

$\left[\begin{array}{ccc}0& 64& -40\\ 9& 11& -11\\ -3& 39& -23\end{array}\right]$