10.5 Homework: multiplying matrices ii

Matrix $\left[A\right]$ is $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ . Matrix $\left[B\right]$ is $\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$ .

• A

  Find the product $\mathrm{AB}$ .

• B

  Find the product $\mathrm{BA}$ .

Multiply.

$\left[\begin{array}{ccc}2& 6& 4\\ 9& 5& 8\end{array}\right]\left[\begin{array}{cccc}2& 5& 4& 7\\ 3& 4& 6& 9\\ 8& 4& 2& 0\end{array}\right]$

Multiply.

$\left[\begin{array}{cccc}2& 5& 4& 7\\ 3& 4& 6& 9\\ 8& 4& 2& 0\end{array}\right]\left[\begin{array}{ccc}2& 6& 4\\ 9& 5& 8\end{array}\right]$

$\left[\begin{array}{ccc}5& 3& 9\\ 7& 5& 3\\ 2& 7& 5\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

• A

  Multiply.

• B

  Now, multiply $\left[\begin{array}{ccc}5& 3& 9\\ 7& 5& 3\\ 2& 7& 5\end{array}\right]\left[\begin{array}{c}2\\ \text{10}\\ 5\end{array}\right]$ —but not by manually multiplying it out! Instead, plug $x=2$ , $y=10$ , and $z=5$ into the formula you came up with in part (a).

Multiply.

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$3\left[\begin{array}{cc}3& -2\\ 6& 3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}9\\ -3\end{array}\right]$

• A

  Find the $x$ and $y$ values that will make this matrix equation true.

• B

  Test your answer by doing the multiplication to make sure it works out.

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}\text{Some}& \text{Matrix}\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

• A

  Find the “some matrix” that will make this matrix equation true.

• B

  Test your answer by doing the multiplication to make sure it works out.