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# Matrix Algebra

## Add, subtract, and multiply matrices based on the positions of entries.

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Matrix Operations with Technology

The local movie theater is running a special for the Martin Luther King Jr. holiday weekend. All matinee tickets are 5, regardless of age. The following matrix shows the number of each type of matinee ticket the movie theater sold over the Martin Luther King, Jr. holiday weekend. Tickets Sold Kids \ Adults \ Seniors \ Sat.Sun. Mon. 1228957133753815010149\begin{align*}& \quad Sat. \quad Sun. \quad \ Mon.\qquad \quad \quad \ \\ \begin{matrix} \text{Kids \ }\\ \text{Adults \ }\\ \text{Seniors \ }\end{matrix} & \begin{bmatrix} 122 & 133 & 150\\ 89 & 75 & 101\\ 57 & 38 & 49\end{bmatrix}\end{align*} How much total money did the movie theater take in for the three-day weekend from kids' ticket sales? ### Matrix Operations To use the calculator to perform matrix operations, students must first learn how to put matrices into their calculator. If you are using a TI-83 or TI-84, access the Matrix menu by either pressing MATRIX or (\begin{align*}2^{nd} \ x^{-1}\end{align*} MATRIX). Now you can choose to EDIT matrix \begin{align*}A\end{align*}. Change the dimensions as needed and enter the data values. You can enter values for a second matrix by selecting matrix \begin{align*}B\end{align*} in the EDIT menu and repeating the process. Once the matrices have been entered, they can be called up to put in an equation on the calculator by choosing the matrix in the NAME menu. If you do not have a TI-83 or 84, the commands might be different. Check with your teacher. Let's multiply the following matrices together. 1. \begin{align*}\frac{2}{3}\begin{bmatrix} 5 & 6\\ -9 & 2\end{bmatrix}\end{align*} This is an example of scalar multiplication. Here we can multiply each of the elements of the matrix by \begin{align*}\frac{2}{3}\end{align*} by hand or we can use the calculator. To use the calculator, we must first enter the matrix. Access the MATRIX menu (press \begin{align*}2^{nd} \ x^{-1}\end{align*} ), select EDIT and matrix \begin{align*}A\end{align*}. Make the dimensions of the matrix \begin{align*}2 \times 2\end{align*} and then enter the values. Now, exit (\begin{align*}2^{nd}\end{align*} QUIT) to return to the home screen. We can now type in our expression by typing \begin{align*}\frac{2}{3}[A]\end{align*} (you can call up matrix \begin{align*}A\end{align*} in the matrix menu by selecting NAME \begin{align*}[A]\end{align*} and ENTER) and press ENTER to get the result. The result can then be put into fraction form by pressing MATH and selecting FRAC. The final reduced fraction form is \begin{align*}\begin{bmatrix} \frac{10}{3} & 4\\-6 & \frac{4}{3}\end{bmatrix}.\end{align*} 1. \begin{align*}\begin{bmatrix} 2 & -3 & 5 & 1\\ 0 & -4 & 3 & -2\\ 7 & 1 & 0 & -5\end{bmatrix}\cdot\begin{bmatrix} 8 & 2 & -1\\ -3 & 0 & 7\\ 5 & -1 & 3\\ 1 & 4 & 6\end{bmatrix}\end{align*} This multiplication problem would be very tedious to perform by hand. We can use the calculator to make the process much more efficient and avoid making careless arithmetic errors as well. Go to the MATRIX menu on your calculator and EDIT matrix \begin{align*}A\end{align*}. Note that the dimensions should be \begin{align*}3 \times 4\end{align*}. Now, enter the elements of the matrix. When you are finished, exit (\begin{align*}2^{nd}\end{align*} QUIT) to return to the home screen. Return to the MATRIX menu and EDIT matrix \begin{align*}B\end{align*}. Note that the dimensions of matrix \begin{align*}B\end{align*} are \begin{align*}4 \times 3\end{align*}. Enter the data elements and return to the home screen. Now we can multiply the matrices together by calling them up from the MATRIX menu. Return to the MATRIX menu and select NAME \begin{align*}[A]\end{align*}. You will be automatically returned to the home where \begin{align*}[A]\end{align*} should be showing. Now, return to the MATRIX menu and select NAME \begin{align*}[B]\end{align*}. Your home screen should now have \begin{align*}[A][B]\end{align*} . It is not necessary to put a multiplication symbol between the two matrices since the brackets act like parenthesis and already indicate multiplication. Press ENTER and the result will appear: \begin{align*}\begin{bmatrix} 51 & 3 & -2\\ 25 & -11 & -31\\ 48 & -6 & -30\end{bmatrix}.\end{align*} Now, let's evaluate the following expression: \begin{align*}\frac{1}{2}\begin{bmatrix} 8 & -3\\ 5 & 1\end{bmatrix}\cdot\begin{bmatrix} -4 & 1\\ 6 & 7\end{bmatrix} + \begin{bmatrix} -2 & -1\\ 9 & 0\end{bmatrix}\end{align*} This time we will need to enter three matrices into the calculator. Let’s let \begin{align*}[A] = \begin{bmatrix} 8 & -3\\ 5 & 1\end{bmatrix}, [B] = \begin{bmatrix} -4 & 1\\ 6 & 7\end{bmatrix}, [C] = \begin{bmatrix} -2 & -1\\ 9 & 0\end{bmatrix}.\end{align*} Once you have entered these matrices, go to the home screen and type in the expression: \begin{align*}\frac{1}{2}[A][B]+[C]\end{align*} by calling up the matrices from the MATRIX menu as needed. Now press ENTER and MATH FRAC to get the final result: \begin{align*}\begin{bmatrix} -27 & -\frac{15}{2}\\ 2 & 6\end{bmatrix}.\end{align*} ### Examples #### Example 1 Earlier, you were asked to find the total amount of money the movie theater took in for the three day weekend. This is a matrix multiplication problem where the matrix is multiplied by a scalar. Tickets Sold \begin{align*}& \quad Sat. \quad Sun. \quad \ Mon.\qquad \quad \quad \ \\ \begin{matrix} \text{Kids \ }\\ \text{Adults \ }\\ \text{Seniors \ }\end{matrix} 5 \cdot & \begin{bmatrix} 122 & 133 & 150\\ 89 & 75 & 101\\ 57 & 38 & 49\end{bmatrix}\end{align*} Use your calculator to assist you. The resulting matrix is: Tickets Sales () \begin{align*}& \quad Sat. \quad Sun. \quad \ Mon.\qquad \quad \quad \ \\ \begin{matrix} \text{Kids \ }\\ \text{Adults \ }\\ \text{Seniors \ }\end{matrix} \begin{bmatrix} 610 & 665 & 750\\ 445 & 375 & 505\\ 285 & 190 & 245\end{bmatrix}\end{align*}

From this matrix, we can see that the movie theater made $610 from kids' ticket sales on Saturday,$665 on Sunday, and \$750, so the total amount made from kid's ticket sales was \begin{align*}\610 + \665 + \750 = \2025\end{align*}.

Use your calculator to evaluate the following matrix expressions.

#### Example 2\begin{align*}\frac{1}{2}\begin{bmatrix}4 & -3\end{bmatrix}\begin{bmatrix} 8 & 4\\ -6 & -2\end{bmatrix}\end{align*}

Let \begin{align*}[A] = [4 \quad -3]\end{align*} and \begin{align*}[B] = \begin{bmatrix} 8 & 4\\ -6 & -2\end{bmatrix}.\end{align*} Now we can go to the home screen and type in the expression: \begin{align*}\frac{1}{2}[A][B]\end{align*} and press ENTER to get: \begin{align*}[25 \quad 11]\end{align*}.

#### Example 3

\begin{align*}\begin{bmatrix} 1 & -2\\ 3 & -4\end{bmatrix}\left( \begin{bmatrix} 2\\ -5\end{bmatrix}+\begin{bmatrix} -4\\ 8\end{bmatrix} \right)\end{align*}

Let \begin{align*}[A] = \begin{bmatrix} 1 & -2\\ 3 & -4\end{bmatrix}, [B] = \begin{bmatrix} 2\\ -5\end{bmatrix}\end{align*} and \begin{align*}[C] = \begin{bmatrix} -4\\ 8\end{bmatrix}.\end{align*} Return to the home screen and type in the expression: \begin{align*}[A]([B]+[C])\end{align*} and press ENTER to get: ERR: INVALID DIM. We know that it is, in fact, possible to add matrix \begin{align*}B\end{align*} and matrix \begin{align*}C\end{align*} and multiply the result by matrix \begin{align*}A\end{align*} on the left.

\begin{align*}^{*}\end{align*}We need to make sure that we multiply \begin{align*}[A]\end{align*} on the left of \begin{align*}[B]+[C]\end{align*} because multiplication of matrices is not commutative (and because the dimensions would not work for multiplication the other way around because we would have a \begin{align*}2 \times 1\end{align*} multiplied by a \begin{align*}2 \times 2\end{align*}.)

Let’s try breaking this one down into these two steps on the calculator:

Step 1: Add \begin{align*}[B]+[C]\end{align*}

Step 2: Now, call up matrix \begin{align*}A\end{align*} and multiply by the answer from step 1: \begin{align*}[A]\end{align*} Ans (to get Ans, press \begin{align*}2^{nd}\end{align*} and (-) button.) for the result: \begin{align*}\begin{bmatrix} -8\\ -18\end{bmatrix}.\end{align*}

Reason for the initial error: The calculator must be trying to multiply the sum of matrices \begin{align*}B\end{align*} and \begin{align*}C\end{align*} by matrix \begin{align*}A\end{align*} on the right and this product is not possible due to the mismatched dimensions.

#### Example 4\begin{align*}\begin{bmatrix} 2 & 5 & -3\\ 8 & -4 & 2\\ -7 & 0 & 6\end{bmatrix}\begin{bmatrix} 7\\ -3\\ 1\end{bmatrix}+5\begin{bmatrix} 4\\ -8\\ 2\end{bmatrix}\end{align*}

Let \begin{align*}[A] = \begin{bmatrix} 2 & 5 & -3\\ 8 & -4 & 2\\ -7 & 0 & 6\end{bmatrix}, [B] = \begin{bmatrix} 7\\ -3\\ 1\end{bmatrix}\end{align*} and \begin{align*}[C] = \begin{bmatrix} 4\\ -8\\ 2\end{bmatrix}.\end{align*} Now we can return to the home screen and type in the expression: \begin{align*}[A][B]+5[C]\end{align*}. Now press ENTER to get: \begin{align*}\begin{bmatrix} 16\\ 30\\ -33\end{bmatrix}.\end{align*}

### Review

Use a calculator to evaluate the following expressions, if possible.

1. .
\begin{align*}\begin{bmatrix} 2\\ -5\end{bmatrix}\cdot\begin{bmatrix} 3 & 1\end{bmatrix}\end{align*}
1. .
\begin{align*}\begin{bmatrix} 4 & -1 & 2\\ -7 & 9 & 0\\ 5 & -6 & 1\end{bmatrix}\cdot\begin{bmatrix} -2 & 5 & 1 & -3\\ -3 & 9 & 10 & -1\\ 0 & 8 & 11 & 1\end{bmatrix}\end{align*}
1. .
\begin{align*}\frac{1}{3}\begin{bmatrix} 3 & -1 & 8\\ 2 & 0 & 5\end{bmatrix}\cdot\begin{bmatrix} 2\\ -1\\ 7\end{bmatrix}\end{align*}
1. .
\begin{align*}-2\begin{bmatrix} 0 & 1 \\ 2 & 0 \end{bmatrix}\cdot\begin{bmatrix} 2 & -5 \end{bmatrix}\end{align*}
1. .
\begin{align*}\left(\begin{bmatrix}2 & -3 & 7\end{bmatrix}+\begin{bmatrix} -3 & 5 & 1\end{bmatrix}\right)\cdot\begin{bmatrix} -4\\ 8\\ 11\end{bmatrix}\end{align*}
1. .
\begin{align*}\begin{bmatrix} 4\\ -3\end{bmatrix}-\begin{bmatrix} 5 & -2\\ 7 & 11\end{bmatrix}\cdot\begin{bmatrix} -6\\ 8\end{bmatrix}\end{align*}
1. .
\begin{align*}\begin{bmatrix} 2 & -6 & 5 & 7\\ 8 & 11 & -1 & -3\\ -4 & 9 & 10 & 12\end{bmatrix}\cdot\begin{bmatrix} -11 & -6 & 1\\ 7 & 10 & 3\\ 8 & 4 & 5\\ 5 & 2 & 6\end{bmatrix}\end{align*}
1. .
\begin{align*}\begin{bmatrix} 4 & 1 & -2\\ -2 & 1 & 6 \\ -12 & 4 & 10\end{bmatrix}\cdot 4\begin{bmatrix} 3\\ 8\\ -9\end{bmatrix}\end{align*}
1. .
\begin{align*}\frac{1}{5}\begin{bmatrix} 1\\ 3\\ 7\end{bmatrix}\cdot\begin{bmatrix}-2 & 8 & 4\end{bmatrix}\end{align*}
1. .
\begin{align*}\begin{bmatrix} -2 & 1 & 6 & 3\\ 3 & 0 & 12 & 4\\ 7 & -5 & -8 & -12\\ -4 & 11 & 1 & 0\end{bmatrix}\cdot\begin{bmatrix} 5 & -3 & 1 & 6\\ -4 & 2 & 13 & 11\\ 6 & 7 & 5 & 3\\ 10 & -8 & -6 & 4\end{bmatrix}\end{align*}
1. .
\begin{align*}\begin{bmatrix} 2 & -8\\ 12 & 5\\ -3 & 11\end{bmatrix}\cdot\begin{bmatrix} 4 & -6 & 10\\ -2 & 8 & 7\end{bmatrix}\end{align*}
1. .
\begin{align*}\frac{1}{3}\begin{bmatrix} 8\\ -7\end{bmatrix}+\frac{1}{2}\begin{bmatrix} -6 & 3 & -5\\ 2 & 11 & 8\end{bmatrix}\cdot\begin{bmatrix} 2\\ -1\\ 7\end{bmatrix}\end{align*}

For problems 13 and 14, the calculator will give you an error. Can you explain why? Is it an error you can work around or is the problem just not possible? If you can work around the issue, complete the problem.

1. .
\begin{align*}\begin{bmatrix}2 & -3\end{bmatrix} \left( \begin{bmatrix} -1 & 11\\ 3 & -7\end{bmatrix}+\begin{bmatrix} 4 & -2\\ 0 & 9\end{bmatrix} \right)\end{align*}
1. .
\begin{align*}\left( \begin{bmatrix} 2\\ -5\end{bmatrix}+\begin{bmatrix} -3\\ 7\end{bmatrix} \right)\begin{bmatrix} -6 & 1\\ 4 & 8\end{bmatrix}\end{align*}
1. Real world application: A movie theater tracks the sales of popcorn in three different sizes. The data collected over a weekend (Friday, Saturday and Sunday nights) is shown in the matrix below. The price of each size is shown in a second matrix. How much revenue did the theater take in each night for popcorn? How much did the theater take in for popcorn in total?
2. \begin{align*}& \quad S \quad M \quad \ L \qquad \quad \quad \ \text{Price}\\ \begin{matrix} \text{Friday \ }\\ \text{Saturday \ }\\ \text{Sunday \ }\end{matrix} & \begin{bmatrix} 36 & 85 & 40\\ 41 & 112 & 51\\ 28 & 72 & 35\end{bmatrix} \qquad \begin{matrix} S\\ M\\ L\end{matrix} \ \begin{bmatrix} 5.50\\ 6.25\\ 7.25\end{bmatrix}\end{align*}

To see the Review answers, open this PDF file and look for section 4.6.

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### Vocabulary Language: English

TermDefinition
Matrices Matrices are rectangular arrangements of data elements presented in rows and columns.
Matrix A matrix is a rectangular arrangement of data elements presented in rows and columns.
matrix operations The primary matrix operations are addition, subtraction and multiplication.
order The order of a matrix describes the number of rows and the number of columns in the matrix.