## M1 - Multiplying matrices

#### Example 1 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & 1 \\ -5 & -4 \\ 0 & -5 \end{array}\right] & & B= \left[\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 5 & 1 & -1 & 3 \\ 1 & 1 & 4 & 4 \end{array}\right] & & C= \left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ -1 & -3 \\ 0 & -4 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BC= \left[\begin{array}{cc} 2 & 5 \\ 6 & 2 \\ -3 & -25 \end{array}\right]$

#### Example 2 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 0 & 2 & -1 & 5 \\ 1 & 0 & -2 & 4 \\ -1 & 3 & 0 & 4 \end{array}\right] & & B= \left[\begin{array}{cc} -3 & -3 \\ -2 & -3 \\ -5 & -6 \\ 4 & 5 \end{array}\right] & & C= \left[\begin{array}{cc} 1 & -1 \\ 3 & -2 \\ 0 & 3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AB= \left[\begin{array}{cc} 21 & 25 \\ 23 & 29 \\ 13 & 14 \end{array}\right]$

#### Example 3 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 1 & 0 & -6 & -3 \\ 0 & 1 & -2 & 4 \\ 0 & -1 & 2 & -3 \end{array}\right] & & B= \left[\begin{array}{cccc} 0 & 1 & 2 & 1 \\ -1 & -2 & -2 & -1 \end{array}\right] & & C= \left[\begin{array}{cc} 0 & -1 \\ 1 & 5 \\ 0 & 1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CB= \left[\begin{array}{cccc} 1 & 2 & 2 & 1 \\ -5 & -9 & -8 & -4 \\ -1 & -2 & -2 & -1 \end{array}\right]$

#### Example 4 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} -1 & 0 & 1 & 2 \\ -2 & -1 & 3 & 6 \end{array}\right] & & B= \left[\begin{array}{cccc} 0 & -1 & -4 & -5 \\ 1 & 0 & -1 & -1 \\ 0 & -1 & -3 & -4 \end{array}\right] & & C= \left[\begin{array}{cc} 1 & 1 \\ 0 & 1 \\ 2 & 4 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cccc} -3 & -1 & 4 & 8 \\ -2 & -1 & 3 & 6 \\ -10 & -4 & 14 & 28 \end{array}\right]$

#### Example 5 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} -1 & -1 & -1 \\ 1 & 0 & -1 \end{array}\right] & & B= \left[\begin{array}{cccc} 2 & -6 & -3 & 2 \\ -1 & 3 & 2 & -2 \end{array}\right] & & C= \left[\begin{array}{ccc} -1 & 1 & -2 \\ -2 & -1 & -6 \\ 0 & -3 & -5 \\ -2 & 0 & -6 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BC= \left[\begin{array}{ccc} 6 & 17 & 35 \\ -1 & -10 & -14 \end{array}\right]$

#### Example 6 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 1 & 5 & -2 & -6 \\ 1 & 5 & -1 & -4 \end{array}\right] & & B= \left[\begin{array}{cccc} 0 & 0 & 1 & 1 \\ 1 & 1 & -5 & 0 \\ -1 & 0 & 1 & -3 \end{array}\right] & & C= \left[\begin{array}{cc} 0 & -3 \\ -1 & -1 \\ 1 & 5 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cccc} -3 & -15 & 3 & 12 \\ -2 & -10 & 3 & 10 \\ 6 & 30 & -7 & -26 \end{array}\right]$

#### Example 7 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & 5 & 0 \\ -1 & -4 & 0 \end{array}\right] & & B= \left[\begin{array}{ccc} 4 & 5 & -3 \\ -4 & -3 & 6 \\ -3 & -4 & 2 \\ -2 & -5 & -2 \end{array}\right] & & C= \left[\begin{array}{cc} 6 & 4 \\ 3 & 1 \\ -5 & -4 \\ -5 & -1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{ccc} 2 & 14 & 0 \\ 2 & 11 & 0 \\ -1 & -9 & 0 \\ -4 & -21 & 0 \end{array}\right]$

#### Example 8 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} -2 & 4 \\ 2 & -3 \\ 3 & -6 \end{array}\right] & & B= \left[\begin{array}{cccc} -1 & 4 & 1 & 3 \\ 1 & 1 & 1 & 6 \\ -1 & 2 & 0 & -1 \end{array}\right] & & C= \left[\begin{array}{cc} 3 & 2 \\ 1 & 1 \\ 2 & 1 \\ -4 & 0 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BC= \left[\begin{array}{cc} -9 & 3 \\ -18 & 4 \\ 3 & 0 \end{array}\right]$

#### Example 9 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & -5 \\ 1 & -4 \\ -1 & 0 \end{array}\right] & & B= \left[\begin{array}{cccc} 1 & 1 & 2 & 1 \\ -3 & -2 & -2 & 3 \\ 1 & 0 & -1 & -3 \end{array}\right] & & C= \left[\begin{array}{cccc} 0 & 1 & -3 & -1 \\ -1 & -1 & 1 & 3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AC= \left[\begin{array}{cccc} 5 & 6 & -8 & -16 \\ 4 & 5 & -7 & -13 \\ 0 & -1 & 3 & 1 \end{array}\right]$

#### Example 10 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & -1 & -2 \\ 4 & -3 & -3 \\ -1 & 1 & 3 \\ -1 & 1 & -3 \end{array}\right] & & B= \left[\begin{array}{cccc} 4 & -3 & -5 & 2 \\ 3 & -2 & -4 & 1 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & 4 & -5 \\ 0 & 1 & -1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{ccc} -5 & 2 & -20 \\ -2 & 0 & -15 \end{array}\right]$

#### Example 11 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & -5 & -5 \\ 0 & 0 & 1 \end{array}\right] & & B= \left[\begin{array}{ccc} 1 & 0 & -1 \\ -5 & 1 & 0 \\ 5 & -1 & 1 \\ 3 & 1 & -5 \end{array}\right] & & C= \left[\begin{array}{cccc} 1 & -1 & -2 & -3 \\ 0 & 1 & 3 & 2 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CB= \left[\begin{array}{ccc} -13 & -2 & 12 \\ 16 & 0 & -7 \end{array}\right]$

#### Example 12 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} -3 & 4 & 4 \\ 2 & -3 & -4 \\ 0 & 1 & 5 \\ 0 & 0 & 1 \end{array}\right] & & B= \left[\begin{array}{cccc} 1 & -1 & 2 & -2 \\ -1 & 2 & -3 & 5 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & 5 & 2 \\ -1 & -4 & -2 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{ccc} -5 & 9 & 16 \\ 7 & -13 & -22 \end{array}\right]$

#### Example 13 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 5 & 6 \\ 1 & -1 & 4 \\ -2 & -5 & -2 \end{array}\right] & & B= \left[\begin{array}{cc} 1 & -1 \\ -3 & 4 \\ -1 & 5 \\ -3 & 0 \end{array}\right] & & C= \left[\begin{array}{cc} 1 & -4 \\ 0 & 1 \\ -3 & 1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AC= \left[\begin{array}{cc} -8 & 1 \\ -16 & 3 \\ -11 & -1 \\ 4 & 1 \end{array}\right]$

#### Example 14 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & 2 \\ 1 & 3 \\ 1 & 5 \\ 0 & -3 \end{array}\right] & & B= \left[\begin{array}{cccc} 1 & 1 & -1 & 1 \\ -4 & -3 & 2 & 0 \\ 0 & 1 & -1 & 1 \end{array}\right] & & C= \left[\begin{array}{cc} -2 & 2 \\ 0 & 1 \\ 3 & -4 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{cc} 1 & -3 \\ -5 & -7 \\ 0 & -5 \end{array}\right]$

#### Example 15 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} -1 & -5 \\ 1 & 4 \\ 3 & 5 \\ 2 & 5 \end{array}\right] & & B= \left[\begin{array}{cc} -2 & -5 \\ 2 & -3 \\ 1 & 1 \end{array}\right] & & C= \left[\begin{array}{cccc} 2 & 1 & -6 & 0 \\ 2 & 1 & -6 & -1 \\ 1 & 1 & -5 & 3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cc} -19 & -36 \\ -21 & -41 \\ -9 & -11 \end{array}\right]$

#### Example 16 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 4 & 3 & 6 & -5 \\ -3 & -2 & -4 & 4 \end{array}\right] & & B= \left[\begin{array}{ccc} -1 & -1 & -1 \\ 2 & 1 & 4 \end{array}\right] & & C= \left[\begin{array}{ccc} 0 & -1 & -3 \\ 2 & -1 & -6 \\ -1 & 0 & 2 \\ 0 & -1 & -4 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AC= \left[\begin{array}{ccc} 0 & -2 & 2 \\ 0 & 1 & -3 \end{array}\right]$

#### Example 17 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 1 & -1 & 5 & 6 \\ 2 & 1 & 2 & -5 \\ 0 & -1 & 3 & 6 \end{array}\right] & & B= \left[\begin{array}{cc} 1 & 4 \\ 0 & 1 \\ -1 & -1 \end{array}\right] & & C= \left[\begin{array}{cccc} -2 & 1 & 0 & -4 \\ -1 & 0 & 1 & -3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BC= \left[\begin{array}{cccc} -6 & 1 & 4 & -16 \\ -1 & 0 & 1 & -3 \\ 3 & -1 & -1 & 7 \end{array}\right]$

#### Example 18 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & -2 & -3 \\ 0 & 1 & 4 \\ 2 & -3 & -5 \\ 0 & 0 & 2 \end{array}\right] & & B= \left[\begin{array}{ccc} 1 & -4 & -2 \\ 0 & 1 & 1 \end{array}\right] & & C= \left[\begin{array}{cccc} 0 & 0 & 1 & 1 \\ -1 & -3 & 2 & 2 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{ccc} 2 & -3 & -3 \\ 3 & -7 & -15 \end{array}\right]$

#### Example 19 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} -1 & 1 \\ 1 & -4 \\ -1 & 5 \\ 2 & -6 \end{array}\right] & & B= \left[\begin{array}{ccc} 3 & -3 & 5 \\ 1 & -2 & 2 \\ 2 & -4 & 5 \\ -1 & 0 & 0 \end{array}\right] & & C= \left[\begin{array}{cc} -1 & 1 \\ -1 & 6 \\ 1 & -3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BC= \left[\begin{array}{cc} 5 & -30 \\ 3 & -17 \\ 7 & -37 \\ 1 & -1 \end{array}\right]$

#### Example 20 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} -2 & -1 & -1 & 0 \\ 1 & 0 & -1 & 2 \\ 2 & -1 & -4 & 6 \end{array}\right] & & B= \left[\begin{array}{cccc} -2 & -1 & -3 & -1 \\ -3 & -2 & -5 & -1 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & 5 & -4 \\ 1 & 6 & -5 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cccc} -5 & 3 & 10 & -14 \\ -6 & 4 & 13 & -18 \end{array}\right]$

#### Example 21 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 1 & 1 & 0 & 2 \\ -3 & -2 & 1 & -5 \end{array}\right] & & B= \left[\begin{array}{ccc} 0 & 1 & 2 \\ -1 & 2 & 5 \end{array}\right] & & C= \left[\begin{array}{cccc} 0 & 1 & 2 & 0 \\ -1 & 4 & 6 & -3 \\ 0 & 2 & 5 & 1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BC= \left[\begin{array}{cccc} -1 & 8 & 16 & -1 \\ -2 & 17 & 35 & -1 \end{array}\right]$

#### Example 22 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & -3 \\ 0 & 1 \\ -1 & 0 \\ 0 & -2 \end{array}\right] & & B= \left[\begin{array}{ccc} -2 & -3 & 0 \\ -1 & -2 & 0 \\ -2 & -5 & 1 \\ 1 & 4 & 3 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & 2 & 5 \\ -1 & -1 & -3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AC= \left[\begin{array}{ccc} 4 & 5 & 14 \\ -1 & -1 & -3 \\ -1 & -2 & -5 \\ 2 & 2 & 6 \end{array}\right]$

#### Example 23 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} -3 & 2 & -3 & 2 \\ 3 & -2 & 4 & -1 \\ -4 & 3 & -6 & 1 \end{array}\right] & & B= \left[\begin{array}{cccc} 1 & 3 & -5 & -6 \\ 0 & 0 & 1 & 1 \end{array}\right] & & C= \left[\begin{array}{ccc} -1 & 3 & 2 \\ 1 & -4 & -4 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cccc} 4 & -2 & 3 & -3 \\ 1 & -2 & 5 & 2 \end{array}\right]$

#### Example 24 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} -1 & -6 \\ 1 & 5 \\ 0 & 0 \\ 0 & -5 \end{array}\right] & & B= \left[\begin{array}{ccc} 6 & -5 & -4 \\ 5 & -4 & -3 \end{array}\right] & & C= \left[\begin{array}{ccc} 4 & 2 & -3 \\ -2 & -3 & -3 \\ -1 & -2 & -2 \\ 1 & 0 & -2 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AB= \left[\begin{array}{ccc} -36 & 29 & 22 \\ 31 & -25 & -19 \\ 0 & 0 & 0 \\ -25 & 20 & 15 \end{array}\right]$

#### Example 25 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 0 & 1 & -1 & -1 \\ -1 & 4 & 0 & -3 \end{array}\right] & & B= \left[\begin{array}{cccc} -2 & -2 & 3 & 2 \\ 1 & 2 & 0 & 2 \\ 3 & 3 & -5 & -4 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & -3 & -3 \\ 2 & -5 & -6 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CB= \left[\begin{array}{cccc} -14 & -17 & 18 & 8 \\ -27 & -32 & 36 & 18 \end{array}\right]$

#### Example 26 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & -1 \\ -2 & 3 \\ 4 & -3 \end{array}\right] & & B= \left[\begin{array}{cc} -1 & 0 \\ -2 & -1 \\ -3 & -4 \\ -5 & -5 \end{array}\right] & & C= \left[\begin{array}{ccc} 0 & 0 & -1 \\ 1 & 0 & -5 \\ 1 & -1 & 1 \\ 2 & -1 & 0 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cc} -4 & 3 \\ -19 & 14 \\ 7 & -7 \\ 4 & -5 \end{array}\right]$

#### Example 27 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 1 & 0 & 1 & 5 \\ 0 & 1 & -3 & -6 \\ 0 & 0 & 1 & 2 \end{array}\right] & & B= \left[\begin{array}{cc} 0 & -1 \\ 1 & -4 \\ 1 & -1 \end{array}\right] & & C= \left[\begin{array}{cccc} 1 & 5 & 3 & -3 \\ -1 & -4 & -3 & 2 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BC= \left[\begin{array}{cccc} 1 & 4 & 3 & -2 \\ 5 & 21 & 15 & -11 \\ 2 & 9 & 6 & -5 \end{array}\right]$

#### Example 28 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 0 & 1 & 1 \\ -1 & -2 & -3 \end{array}\right] & & B= \left[\begin{array}{cccc} -1 & -1 & -1 & 2 \\ 1 & 0 & 1 & -3 \\ 1 & 1 & 2 & -2 \end{array}\right] & & C= \left[\begin{array}{cccc} -1 & -6 & -6 & -4 \\ 1 & 5 & 5 & 3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AB= \left[\begin{array}{cccc} 2 & 1 & 3 & -5 \\ -4 & -2 & -7 & 10 \end{array}\right]$

#### Example 29 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & -1 \\ 0 & 1 \\ 0 & -5 \\ 1 & 0 \end{array}\right] & & B= \left[\begin{array}{ccc} 0 & -1 & -1 \\ 1 & -1 & 3 \\ -2 & 6 & -1 \\ 1 & -5 & -4 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & -3 & -5 \\ 1 & -2 & -4 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AC= \left[\begin{array}{ccc} 0 & -1 & -1 \\ 1 & -2 & -4 \\ -5 & 10 & 20 \\ 1 & -3 & -5 \end{array}\right]$

#### Example 30 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} -2 & -1 & 1 \\ -1 & -1 & 2 \end{array}\right] & & B= \left[\begin{array}{ccc} -1 & 2 & -3 \\ -1 & 1 & -3 \\ -1 & 3 & -2 \\ 0 & 4 & -2 \end{array}\right] & & C= \left[\begin{array}{cccc} 1 & -3 & -1 & -1 \\ 0 & 0 & 1 & 3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CB= \left[\begin{array}{ccc} 3 & -8 & 10 \\ -1 & 15 & -8 \end{array}\right]$

#### Example 31 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 0 & -3 \end{array}\right] & & B= \left[\begin{array}{ccc} 2 & 1 & -5 \\ 0 & 1 & 3 \\ 0 & -2 & -5 \\ -1 & -2 & 0 \end{array}\right] & & C= \left[\begin{array}{cc} -1 & 1 \\ 1 & 5 \\ 1 & 6 \\ -2 & -3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{cc} 2 & 20 \\ 0 & -8 \\ 0 & 13 \\ -1 & -4 \end{array}\right]$

#### Example 32 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & 0 & 1 \\ -1 & 1 & 1 \end{array}\right] & & B= \left[\begin{array}{ccc} 4 & 1 & 4 \\ -1 & 0 & 0 \\ -1 & 0 & 1 \\ 0 & -1 & -5 \end{array}\right] & & C= \left[\begin{array}{cc} 3 & -5 \\ 2 & -3 \\ 1 & -6 \\ 3 & -2 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{ccc} 8 & -5 & -2 \\ 5 & -3 & -1 \\ 7 & -6 & -5 \\ 5 & -2 & 1 \end{array}\right]$

#### Example 33 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 1 & 3 & 5 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & -2 & -3 & -5 \end{array}\right] & & B= \left[\begin{array}{ccc} -1 & -3 & 6 \\ 1 & 2 & -5 \end{array}\right] & & C= \left[\begin{array}{cccc} -1 & 1 & -1 & 1 \\ 2 & -3 & 4 & -5 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{cccc} -1 & -18 & -29 & -48 \\ 1 & 15 & 24 & 39 \end{array}\right]$

#### Example 34 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & -2 \\ -1 & 3 \\ 3 & -3 \end{array}\right] & & B= \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ -1 & 3 \\ 4 & -6 \end{array}\right] & & C= \left[\begin{array}{ccc} 4 & 6 & -5 \\ 0 & 1 & 2 \\ -3 & -5 & 3 \\ 2 & 6 & 1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cc} -17 & 25 \\ 5 & -3 \\ 11 & -18 \\ -1 & 11 \end{array}\right]$

#### Example 35 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & -2 \\ 0 & 1 \\ 1 & -1 \end{array}\right] & & B= \left[\begin{array}{cc} 1 & 1 \\ 1 & 2 \\ -2 & -4 \\ -2 & 0 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 5 \\ -1 & 2 & 6 \\ 0 & 1 & 0 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cc} 1 & -3 \\ 5 & -4 \\ 5 & -2 \\ 0 & 1 \end{array}\right]$

#### Example 36 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & 0 & -1 \\ 3 & 1 & -6 \\ -1 & -1 & 5 \\ -1 & -1 & 6 \end{array}\right] & & B= \left[\begin{array}{cccc} 1 & 1 & 1 & -2 \\ 0 & 1 & 2 & -1 \end{array}\right] & & C= \left[\begin{array}{ccc} -3 & 2 & 1 \\ -2 & 1 & 0 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{ccc} 5 & 2 & -14 \\ 2 & 0 & -2 \end{array}\right]$

#### Example 37 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} -4 & 5 & 1 \\ -5 & 6 & 2 \\ 2 & -2 & -1 \\ 4 & -5 & 0 \end{array}\right] & & B= \left[\begin{array}{cc} 1 & 4 \\ 1 & 5 \\ -1 & -6 \\ 0 & -4 \end{array}\right] & & C= \left[\begin{array}{cc} 2 & 5 \\ -1 & -2 \\ -1 & 1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AC= \left[\begin{array}{cc} -14 & -29 \\ -18 & -35 \\ 7 & 13 \\ 13 & 30 \end{array}\right]$

#### Example 38 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} -1 & 0 \\ 1 & -3 \\ -2 & 4 \end{array}\right] & & B= \left[\begin{array}{cc} 1 & 4 \\ 1 & 6 \\ 0 & 1 \\ -1 & -4 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -3 \\ -4 & 3 & -4 \\ 1 & -3 & 6 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{cc} 1 & -4 \\ 7 & -15 \\ 15 & -25 \\ -16 & 33 \end{array}\right]$

#### Example 39 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} -1 & 1 & 1 & 1 \\ -1 & 0 & 2 & 1 \end{array}\right] & & B= \left[\begin{array}{cc} -1 & -3 \\ 2 & 5 \\ 2 & 3 \end{array}\right] & & C= \left[\begin{array}{cccc} 1 & 4 & 5 & -3 \\ 0 & 1 & 1 & 0 \\ 0 & 3 & 4 & -1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{cccc} 4 & -1 & -7 & -4 \\ -7 & 2 & 12 & 7 \\ -5 & 2 & 8 & 5 \end{array}\right]$

#### Example 40 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} -5 & -3 & -2 \\ -3 & -2 & -2 \\ 0 & 1 & 5 \\ -4 & -4 & -5 \end{array}\right] & & B= \left[\begin{array}{cccc} -2 & 1 & -4 & 0 \\ 1 & -1 & 3 & -1 \end{array}\right] & & C= \left[\begin{array}{ccc} -2 & 3 & -5 \\ -3 & 4 & -6 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{ccc} 7 & 0 & -18 \\ 2 & 6 & 20 \end{array}\right]$

#### Example 41 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cccc} 2 & -1 & 5 & 2 \\ 1 & 0 & 2 & 1 \end{array}\right] & & B= \left[\begin{array}{cccc} 1 & -1 & -5 & -3 \\ 0 & 1 & 0 & -3 \\ -1 & 2 & 6 & 1 \end{array}\right] & & C= \left[\begin{array}{ccc} -1 & -3 & 2 \\ -1 & -4 & 2 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CB= \left[\begin{array}{cccc} -3 & 2 & 17 & 14 \\ -3 & 1 & 17 & 17 \end{array}\right]$

#### Example 42 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & 4 & 4 \\ 1 & 5 & 5 \end{array}\right] & & B= \left[\begin{array}{cccc} 0 & 0 & -1 & 1 \\ 1 & -4 & -3 & 3 \\ 1 & -5 & -2 & 2 \end{array}\right] & & C= \left[\begin{array}{cccc} 0 & 1 & 0 & -1 \\ -1 & 6 & 2 & -3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AB= \left[\begin{array}{cccc} 8 & -36 & -21 & 21 \\ 10 & -45 & -26 & 26 \end{array}\right]$

#### Example 43 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} -1 & -3 \\ 1 & 2 \\ -4 & -2 \end{array}\right] & & B= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ -1 & 5 & 6 & 2 \end{array}\right] & & C= \left[\begin{array}{cccc} 1 & -1 & 1 & -3 \\ 1 & 0 & -1 & 2 \\ -1 & 2 & -2 & 5 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AB= \left[\begin{array}{cccc} 3 & -15 & -19 & -6 \\ -2 & 10 & 13 & 4 \\ 2 & -10 & -16 & -4 \end{array}\right]$

#### Example 44 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} -5 & -2 \\ -5 & -4 \\ -2 & -1 \\ 5 & 1 \end{array}\right] & & B= \left[\begin{array}{ccc} 1 & 4 & 5 \\ -1 & -2 & -1 \\ 0 & 1 & 4 \\ 0 & 1 & 1 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & 0 & 3 \\ -1 & 1 & -5 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AC= \left[\begin{array}{ccc} -3 & -2 & -5 \\ -1 & -4 & 5 \\ -1 & -1 & -1 \\ 4 & 1 & 10 \end{array}\right]$

#### Example 45 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{cc} 1 & 2 \\ -3 & -5 \\ 4 & 4 \\ 4 & 5 \end{array}\right] & & B= \left[\begin{array}{cc} -3 & -1 \\ 0 & 1 \\ 2 & 2 \end{array}\right] & & C= \left[\begin{array}{ccc} 6 & 0 & 1 \\ -1 & 1 & 4 \\ 1 & -1 & -3 \\ -5 & 0 & -1 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CB= \left[\begin{array}{cc} -16 & -4 \\ 11 & 10 \\ -9 & -8 \\ 13 & 3 \end{array}\right]$

#### Example 46 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 1 & -5 & -3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right] & & B= \left[\begin{array}{ccc} 1 & 2 & 3 \\ -1 & -1 & -2 \end{array}\right] & & C= \left[\begin{array}{cccc} 1 & 1 & 0 & -5 \\ 0 & 1 & -2 & -3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$CA= \left[\begin{array}{ccc} 1 & -4 & -1 \\ 0 & 1 & 0 \end{array}\right]$

#### Example 47 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} 3 & 5 & 6 \\ 1 & 2 & 3 \\ -1 & -1 & 1 \\ 3 & 2 & -1 \end{array}\right] & & B= \left[\begin{array}{cc} 0 & 4 \\ -2 & -5 \\ 1 & -1 \end{array}\right] & & C= \left[\begin{array}{cc} 4 & 1 \\ -5 & 1 \\ 3 & 1 \\ 3 & -3 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$AB= \left[\begin{array}{cc} -4 & -19 \\ -1 & -9 \\ 3 & 0 \\ -5 & 3 \end{array}\right]$

#### Example 48 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} -5 & -2 & 3 \\ -2 & -1 & 2 \\ 1 & 1 & -2 \\ 1 & 0 & 1 \end{array}\right] & & B= \left[\begin{array}{cc} 1 & -4 \\ 1 & -3 \\ 2 & -3 \\ 2 & -3 \end{array}\right] & & C= \left[\begin{array}{ccc} -2 & -1 & 4 \\ 3 & 1 & -6 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BC= \left[\begin{array}{ccc} -14 & -5 & 28 \\ -11 & -4 & 22 \\ -13 & -5 & 26 \\ -13 & -5 & 26 \end{array}\right]$

#### Example 49 🔗

Of the following three matrices, only two may be multiplied. \begin{align*} A= \left[\begin{array}{ccc} -1 & -4 & 5 \\ -1 & -4 & 4 \end{array}\right] & & B= \left[\begin{array}{cc} 4 & -3 \\ -2 & 5 \\ -3 & 1 \\ 1 & -4 \end{array}\right] & & C= \left[\begin{array}{ccc} 1 & 0 & -2 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \\ 3 & 0 & -5 \end{array}\right] \\ \end{align*} Explain which two can be multiplied and why. Then show how to find their product.

$BA= \left[\begin{array}{ccc} -1 & -4 & 8 \\ -3 & -12 & 10 \\ 2 & 8 & -11 \\ 3 & 12 & -11 \end{array}\right]$
$BC= \left[\begin{array}{cc} 1 & -28 \\ 2 & 23 \\ -1 & -8 \end{array}\right]$