How to Solve Matrices (SAT, ACT, etc.)

Given that the ACT is on July 18th, 2020 and the upcoming SAT is on August 29th, 2020, I noticed that many students are confused about what matrices are.

Don’t worry! In this post, I will cover what a matrix is from scratch, what type of problems you can expect on the tests, and even give you some practice problems!

What are matrices?

So what are matrices? Honestly, it isn’t too hard to understand. A matrix is simply a table of numbers, as shown below.

$\begin{bmatrix} 1 & 0 & 2 & 3 \\ 1 & 9 & -2 & 3 \\ 2.3 & \pi & 7.7 & -3.4 \\ 191.26 & 11 & -2 & 0 \end{bmatrix}$

Generally, we put brackets or parentheses around our matrix to indicate that it is in fact a matrix. As you can see above, any number can go into a matrix, like $\pi$ for example.

Aside from the actual values stored in our matrix (or table), the other important factor to consider is its dimensions, namely how many rows and columns it has. Try answering the questions below!

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Matrix Rows and Columns Question

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So, how many rows and columns does the matrix below have?

$\begin{bmatrix} 1 & 0 & 2 & 3 \\ 1 & 9 & -2 & 3 \\ 2.3 & \pi & 7.7 & -3.4 \\ 191.26 & 11 & -2 & 0 \end{bmatrix}$

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Well, if you said 4 rows and 4 columns, then you are right! The reason why the number of rows and columns of a matrix is important is that it tells us how big a matrix is.

Sometimes, people will call the matrix above a 4 by 4 matrix because it has 4 rows and 4 columns. In the same way, a 3 by 2 matrix would have 3 rows and 2 columns.

But yeah, that’s basically a matrix is at its core: a table of numbers.

Just like normal numbers, you can also add and subtract matrices! How, you ask? Simple. Just add corresponding elements as shown below.

$\begin{bmatrix} 1 & 0\\ 1 & 9\end{bmatrix} + \begin{bmatrix} 2 & 7\\ -1 & 0.1 \end {bmatrix} = \begin{bmatrix} 3 & 7\\ 0& 9.1 \end {bmatrix}$

To add the matrices above, all I did was take each element in the first matrix and added it with the corresponding element (the ones in the same position) in the second matrix to get the output elements in the third matrix.

To subtract matrices, you just subtract corresponding elements like with addition.

$\begin{bmatrix} 1 & 0\\ 1 & 9\end{bmatrix} - \begin{bmatrix} 2 & 7\\ -1 & 0.1 \end {bmatrix} = \begin{bmatrix} -1 & -7\\ 2& 8.9 \end {bmatrix}$

Note: If you are adding or subtracting matrices, they have to be the same size, meaning they have to have the SAME NUMBER OF ROWS AND SAME NUMBER OF COLUMNS.

These show up every once in a while on the ACT and the SAT (though not often), show why not try a few questions below.

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$\begin{bmatrix} 1 & 0 & 2 \\ -1 & 9 & 2 \\ -4 & 5 & 2 \end{bmatrix} + \begin{bmatrix} 9 & -1 & 2 \\ 5 & 3 & 1 \\ -9 & 10 & -4 \end{bmatrix} =$

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$\begin{bmatrix} 1 & 0\\ 1 & 9\end{bmatrix} - \begin{bmatrix} 2 & 7\\ -1 & 0.1 \end {bmatrix} =$

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What requirements do we need for matrix addition/subtraction?

The average score is 86%

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Scalar Multiplication

The other thing to know about matrices is that they can be multipled with numbers. We call this scalar multiplication. When we multiply a matrix by a number, we take each element in the matrix and multiply it by the number to get the output number as shown below.

$3\begin{bmatrix} 1 & 0\\ 1 & 9\end {bmatrix} = \begin{bmatrix} 3 & 0\\ 3 & 27\end {bmatrix}$

Why don’t you try a few questions to see if you got it?

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Matrix Scalar Multiplication

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$7\begin{bmatrix} 8 & 1 & 0 \\ 1 & -1 & 1 \end{bmatrix}$

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$-1 \begin{bmatrix} 7\end{bmatrix}$

The average score is 90%

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Multiplying Matrices

This is the last and final type of question on matrices the ACT/SAT normally asks about: Multiplying two matrices. Now bear with me because multiplying matrices can be hard. So instead of actually writing how to do it (because it will be confusing), I will link a video below by Math Meeting who I think explains it pretty well and shows it well.

Note: Matrix multiplication is NOT commutative. They like to test this a lot on the ACT and SAT.

If you think you got it, try some questions!

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Matrix Multiplication

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$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} * \begin{bmatrix} 5 & -1 \\ 3 & 2 \end{bmatrix}$

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$\begin{bmatrix} 7 & 6 & 2 \\1 & 0 & 2 \\ -3 & 0.5 & 4\end{bmatrix} * \begin{bmatrix} -1 & 2 & 3 \\4 & 8 & 7 \\ -3 & 0 & 0\end{bmatrix}$ (Challenge)

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Is matrix multiplication commutative?

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Is matrix division possible?

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What are the requirements for matrix multiplication?

The average score is 52%

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Studying Strategy

Because these tests are coming soon, unless you are already getting pretty high scores on the math section, I would advise studying more common questions. Matrices are not normally common on the SAT and ACT. In fact, some tests don’t even have a question on matrices, while others have at most 2. Regardless, knowing about matrices is always nice because of their widespread applications in Computer Science (handling data, etc.) and Linear Algebra. I will link resources below to help you study more on matrices for these tests. Good luck and subscribe if you want to stay updated!

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