Test Prep

# Advanced ACT Math: Multiplying Matrices

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According to students, matrix problems on the ACT are a major source of dread. And of those, multiplying matrices is the worst. Not only are they an advanced math topic, but they rarely show up more than once per test. This means that a lot of students may not get enough exposure through practice tests to adequately prepare. Now, if you're simply aiming to boost your score, you may want to focus on a more fundamental topic that'll have more questions, such as algebra or geometry. But if you're aiming for a top score, just make sure you follow these steps in order, and your matrix multiplication woes will turn into some free points.

## The Golden Rule

First off, remember this: Unlike normal multiplication, matrix multiplication is not commutative. In other words, if A and B are matrices, then A × B won't necessarily (and in fact typically won't) equal B × A. In fact, there is no guarantee that two matrices can even be multiplied together. In short, order matters, so make sure you're multiplying the right way.

### Step 1: Determine If Your Matrices Can Be Multiplied

In order to multiply matrices, the number of columns of the first matrix must match the number of rows in the second matrix. For example, a 2 x 3 matrix could be the multiplied by a 3 x 2 matrix, but a 3 x 2 matrix could not be multiplied by another 3 x 2 matrix. One easy way to remember this is that the middle two numbers must match up when looking at the dimensions.

So for that first example, [2 x 3][3 x 2], the 3s match in the middle. For the second example, [3 x 2][3 x 2], they do not.

### Step 2: Figure out the Dimensions of the Resulting Matrix

When we multiply the matrices from our first example, the resulting matrix will have the dimensions of the outer numbers. In this case, the [2 x 3][3 x 2] multiplication will result in a 2 x 2 matrix, such as the empty one shown below.

### Step 3: Multiply Piece by Piece

Let's take a look at the following matrices to walk through multiplication. Here the first matrix is a 2x3 matrix (2 rows by 3 columns), and the second is a 3x2 matrix (3 rows by two columns):

Start with the first row of the first matrix (1, 2, 3) and the first column of the second matrix (2, 3, 1). Multiply the corresponding numbers together (the first by the first, second by the second, etc.), and then add the products together: (1*2) + (2*3) + (3*1) = 2 + 6 + 3 = 11. This process is called the "dot product."

Next, repeat this process for the first row (1, 2, 3) and the second column (4, 5, 6): (1*4) + (2*5) + (3*6) = 4 + 10 + 18 = 32. This will go in the next slot for the first row:

Now, find the dot product of the second row (7, 3, 4) and the first column (2, 3, 1): (7*2) + (3*3) + (4*1) = 14 + 9 + 4 = 27.

Lastly, find the dot product of the second row (7, 3, 4) and second column (4, 5, 6): (7*4) + (3*5) + (4*6) = 28 + 15 + 24 = 67.

That's it! Turns out, multiplying matrices is no more than elementary school adding and multiplying, so long as you maintain the correct order of rows and columns. The steps might seem a bit complex at first, but with some practice you'll be able to tackle these questions on the ACT. If you need more practice with ACT problems, check out our book, ACT Prep. Plus, subscribe to our YouTube channel for regularly new content to help you with your test prep and college goals.