Matrices are becoming more and more important on the ACT® Math section. They're widely used in advanced computer science. They're really a special form of adding and multiplying quantities of numbers. There are some special rules to them. Make sure you master them. You'll be able to get some more points on the ACT® Math section.
Before we look at matrix operations, let's quickly define what a matrix is. A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. They're used to represent and solve various mathematical problems efficiently.
Add the following Matrices
| [1, 2] | + | [4, 3] | |
|---|---|---|---|
| [3, 4] | [2, 1] |
Now, let's look at how we add matrices. When we add matrices, we add the corresponding elements in each position.
- Add the elements in the first row and first column: 1 + 4 = 5
- Add the elements in the first row and second column: 2 + 3 = 5
- Add the elements in the second row and first column: 3 + 2 = 5
- Add the elements in the second row and second column: 4 + 1 = 5
That would give us the following result:
| [5, 5] |
|---|
| [5, 5] |
Consider this example
Let's say there's a student, Bethany. She sells slime on TikTok. Because she's a savvy businesswoman, she keeps track of her product inventory quite well. She keeps some at her dad's house, and some at her mom's house. The table below shows her current inventory.
- Mom's house has Red: 20, Yellow: 30, Green: 25
- Dad's house has Red: 15, Yellow: 25, Green: 30
- Cost: Red: $5, Yellow: $10, Green: $15
Suppose Bethany wanted to know the total value of all of her slime. How would she do that? An easy way to do that would be to create a matrix. By multiplying the quantity matrix by the price matrix, we can calculate the total value of slime at each location.
| [20, 30, 25] | * | [5] | ||
|---|---|---|---|---|
| [15, 25, 30] | [10] | |||
| [15] |
The matrix shows the slime at her mom's house and her dad's house. Then it multiplies each color of slime by its respective value. This multiplication represents the total value of slime at each location. Try to figure out what the total value of Bethany's product using the matrix above.
Matrix Multiplication
Slime Inventory:
- Mom's house: Red: 20, Yellow: 30, Green: 25
- Dad's house: Red: 15, Yellow: 25, Green: 30
- Cost per item: Red: 5, Yellow: 10, Green: 15
Step-by-step solution:
Quantities from Mom's and Dad's houses can be represented as a 2x3 matrix:
| [20, 30, 25] |
|---|
| [15, 25, 30] |
Costs can be represented as a 3x1 matrix:
| [5] |
|---|
| [10] |
| [15] |
When we multiply matrices, we're essentially performing a series of multiplications and additions that allow us to combine information from both matrices. In this case, we're using the quantities of each type of slime and their respective prices to calculate the total value.
Step-by-Step Matrix Multiplication:
Step 1: Multiply and sum the elements for Mom's house:
- Red items: 20 * 5 = 100
- Yellow items: 30 * 10 = 300
- Green items: 25 * 15 = 375
Sum the results: 100 + 300 + 375 = 775
Step 2: Multiply and sum the elements for Dad's house:
- Red items: 15 * 5 = 75
- Yellow items: 25 * 10 = 250
- Green items: 30 * 15 = 450
Sum the results: 75 + 250 + 450 = 775
Consider another example
Let's say Bethany wants to sell purple slime. 15 purple slime at her mom's and 20 at her dad's. If we add a column to the matrix for purple slime, it'd look like this:
| [20, 30, 25, 15] | * | [5] | = | [775] |
|---|---|---|---|---|
| [15, 25, 30, 20] | [10] | [775] | ||
| [15] |
If we wanted to figure out the total value of Bethany's product now, how would we do that? You'd use all of the same steps as before, except this time, what would you multiply the 15 and 20 by? Nothing. There's nothing there you can multiply it by. You can't complete the problem.
This illustrates an important rule in matrix multiplication: the number of columns in the first matrix must equal the number of rows in the second matrix. In this case, we've added a fourth column to our quantity matrix (for purple slime), but our price matrix still only has three rows. This mismatch in dimensions makes the multiplication impossible.
That might seem kind of obvious. That's the exact question the ACT® Math asks over and over again. You can't do matrix multiplication when the columns on your first matrix don't match the rows on your second.
Sometimes the number of rows and columns a matrix has is expressed as r × c, where r = rows and c = columns. To represent two matrices being multiplied, then, you could write r × c * r × c. Matrix multiplication is not possible when the first c and second r are not equal.
How is this going to show up on the ACT® Math section?
Well, this can show up in a lot of ways. The ACT® Math might ask you to add two matrices. Equally likely is them asking you to multiply two matrices. Or, given two matrices that are equal, reverse engineer (through algebra) a missing value. They might ask you to determine whether matrix multiplication is even possible given two matrices.
This topic has become increasingly important for the ACT® Math section in the past 5 years. Plan to see some questions on matrices – but don't worry too much. You'll be able to solve these questions with the information on this page. Topics like this one add up: here's how much a focused student can realistically improve their ACT® score.
Further Study
Sometimes the ACT® Math section will simply ask what the determinant of a matrix is. Rarer is the case when they ask you to find the determinant.
The determinant is a special number that can be calculated from a square matrix. It's important because it tells us useful information about the matrix, such as whether the matrix is invertible. Understanding determinants can help solve systems of linear equations and is really important in advanced mathematics.
For any given matrix:
| [a, b] |
|---|
| [c, d] |
Determinant of a 2x2 matrix = ad - bc