In order to multiply two matrices together, the number of columns from the first matrix (leftmost) must be equal to the number of rows from the second matrix (rightmost). Once we have confirmed this fact, we set up a matrix for our product with the number of rows from the first matrix and the number of columns from the second matrix. To obtain a given individual entry, we use the row and column location of that entry. Once we have that information, we use the row to find the row to work within the first matrix (leftmost) and we use the column to find the column to work within the second matrix (rightmost). The given entry is found by obtaining the dot product of that given row by that given column. We can find all entries of the matrix using the same strategy.

Test Objectives
• Demonstrate the ability to multiply matrices
Multiplying Matrices Practice Test:

#1:

Instructions: find AB

$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-5 & 5 & -4\\ -2 & 2 & 2\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-3 & -6\\ 2 & 3 \\ 6 & 1\end{array}\right]$$

$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}4 & -12 & 1\\ 1 & 5 & 0\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-4 & 1 & 9\\ -3 & 7 & -6\end{array}\right]$$

#2:

Instructions: find AB

$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-6 & -3 & -1\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-3 & 6\\ -2 & -6 \\ 6 & -1\end{array}\right]$$

$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}9 & 2 & 7\\ 5 & 8 & -1 \\ 0 & 0 & -3\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}-1 & 2 & 13\\ -3 & 0 & 7 \\ -4 & -1 & 12\end{array}\right]$$

#3:

Instructions: find AB

$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-5 & 0 & 6\\ 5 & -1 & 1 \\ 2 & -1 & -6\end{array}\right]$$ $$B=\left[ \begin{array}{cc}0 & -1\\ -3 & -3 \\ -2 & -4\end{array}\right]$$

Instructions: find ABC

$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}0 & -6 & 5\\ -1 & 2 & -1\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-3 & -5\\ -5 & -1\\ -1 & -3\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}0 & -1 & 2\\ 1 & -2 & 1\end{array}\right]$$

#4:

Instructions: find ABC

$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cccc}-6 & 6 & -3 & 4\\ -5 & 3 & -4 & 6\end{array}\right]$$ $$B=\left[ \begin{array}{cc}4 & -1\\ 0 & 4 \\-2 & -5\end{array}\right]$$ $$C=\left[ \begin{array}{cccc}3 & 0 & -6 & 3\\ -2 & -5 & -6 & 0\end{array}\right]$$

Instructions: find AB

$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}-y & -6x & -2\\ x & xy & 6y\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-5x & -6\\ 2 & 4 \\-3x & x\end{array}\right]$$

#5:

Instructions: find AB

$$a)\hspace{.2em}$$ $$A=\left[ \begin{array}{cc}4x & xy\end{array}\right]$$ $$B=\left[ \begin{array}{ccc}x^2 & 0 & 6xy\\ -6x & 0 & -3\end{array}\right]$$

Instructions: find ABC

$$b)\hspace{.2em}$$ $$A=\left[ \begin{array}{ccc}xy & 4x & -5x\\ x^2 & 2y & 5y\end{array}\right]$$ $$B=\left[ \begin{array}{cc}-2y & 4\\ -3x & -4 \\ -5y & y\end{array}\right]$$ $$C=\left[ \begin{array}{cc}-x & 0\\ 5xy & 5x \\ -3 & 3 \\ -2 & 2y\end{array}\right]$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}1 & 41\\ 22 & 20\end{array}\right]$$

$$b)\hspace{.2em}Doesn't \hspace{.2em}Exist$$

#2:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{c}18 & -17\end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-43 & 11 & 215\\ -25 & 11 & 109 \\ 12 & 3 & -36\end{array}\right]$$

#3:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc}-12 & -19\\ 1 & -6 \\15 & 25\end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}-9 & -7 & 41\\ 6 & -6 & -6\end{array}\right]$$

#4:

Solutions:

$$a)\hspace{.2em}Doesn't \hspace{.2em}Exist$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc}5xy - 6 & -26x + 6y\\ -5x^2 - 16xy & -6x + 10xy\end{array}\right]$$

#5:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc}4x^3 - 6x^2y & 0 & 24x^2y - 3xy\end{array}\right]$$

$$b)\hspace{.2em}Doesn't \hspace{.2em}Exist$$