Matrix Notation, Equal Matrices & Math Operations with Matrices

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  • 0:03 Matrices
  • 0:31 Notation
  • 1:16 Equal Matrices
  • 2:08 Matrix Addition and…
  • 3:02 Matrix Multiplication
  • 6:06 Lesson Summary
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Lesson Transcript
Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

Watch this video lesson to learn about the basics of matrices and what kinds of math operations you can do with them. Also learn how to determine whether two matrices are the same or not.


Matrices are math objects where the numbers are organized into a nice rectangular array of rows and columns. They are useful to learn because we can turn a system of equations into matrix form. We can manipulate matrices more easily than a collection of equations. To get you on the right track, we will look at matrices in this video lesson, their proper notation, their allowed math operations, and how to tell when two matrices are equal to each other.


To begin, we look at the proper notation for matrices. Look at this properly written matrix:

Proper notation for matrices

Our numbers are arranged in neat rows and columns. Surrounding our matrix are square brackets. These square brackets tell us that this particular group of numbers belongs together in one matrix. We can also label our matrix by calling it matrix A. If we have labeled our matrix, we can either write out the matrix with all the numbers in it or we can write it using our label surrounded by square brackets. Our matrices are also described by their size. This particular matrix has three rows and three columns, so we can also call it a 3x3 matrix. A matrix with four rows and two columns is a 4x2 matrix.

Equal Matrices

Okay, now that we know what proper matrix notation looks like, let's talk about when two matrices are equal to each other. You know how when two numbers are the same, they are the identical number, like when you have 2 and 2. Well, matrices are similar, but there is just a bit more involved. When two matrices are the same, all the numbers must be the same in the same positions and the matrices must both be the same size. For example, these two matrices are the same:

Equal matrices

They are both the same size and all the numbers are the same in the same spots. They are both 2x3 matrices.

Take a look at these matrices. Are these the same?

These matrices are not equal.

They are not equal matrices because they are different sizes. Just because two matrices have all the same numbers does not mean they are equal. Their sizes must also be the same.

Matrix Addition and Subtraction

Let's cover the kinds of math operations we can do. We can do addition and subtraction. Matrix addition and subtraction is the same as number addition and subtraction. We add or subtract number by number. Because we have to match our number to number, our two matrices must be the same size, like this:

Matrices must be the same size to add or subtract.

We matched the numbers together. We matched the number in the first row and first column in the first matrix to the number in the first row and first column in the second matrix and so on. We can't add or subtract two matrices that are different sizes. We can however add or subtract the same number to all the numbers in a matrix.

Adding the same number to all the matrix numbers

We perform subtraction in the same way as addition. The two matrices must be the same size and we subtract number to number, matching the location of the numbers together. We can also subtract the same number from all the numbers in a matrix.

Matrix Multiplication

The last math operation we can do is matrix multiplication. There is no such thing as matrix division. Matrix multiplication is more complicated than number multiplication. You can easily multiply 3 and 5 to make 15. But with matrices, when we multiply two matrices together, we have to use a combination of multiplication and addition. Also, the number of columns in the first matrix must match the number of rows in the second matrix. So, we can multiply a 1x3 matrix with a 3x2 matrix, but we can't multiply a 1x3 matrix with a 2x3 matrix. And unlike numbers where the order of multiplication doesn't matter, with matrices, which matrix comes first matters.

Let's look at how we multiply two matrices together:

Multiplication with matrices

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Additional Activities

Additional Matrix Operation Examples

In the following examples, students will apply their knowledge of addition, subtraction, and multiplication of matrices to determine various sums, differences, and products. Some problems will use only one operation whereas others will combine operations.


For the following matrices,

determine the following, if they exist.

1) A + E

2) B - C

3) DE

4) ED

5) (A + E)B

6) (A - E)C


1) Since A and E are the same size (they are both 3x2 matrices), we can add them by adding their corresponding elements.

2) We cannot calculate B - C because the matrices are not the same size.

3) We can calculate DE because the number of columns of D matches the number of rows of E.

Remember to multiply a row of D by a column of E. For example, the first entry in the product is found by multiplying the first row of D by the first column of E to get 3(2) + 2(0) + 6(1) = 6 + 0 + 6 = 12.

4) We cannot calculate ED because the number of columns of E (2) does not match the number of rows of D (3). Note that this means that the order of multiplication matters - we could calculate DE but not ED.

5) We found A + E in number 1. We can multiply (A + E) and B because the number of columns of (A + E) matches the number of rows of B.

6) We can subtract E from A because the matrices are the same size. Then, we can multiply (A - E) by C because the number of columns of (A - E) matches the number of rows of C.

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