Square Matrix: Definition & Concept

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  • 0:00 What Is A Square Matrix?
  • 0:54 The Identity Matrix
  • 1:34 Adding Square Matrices
  • 2:22 Multiplying Square Matrices
  • 4:03 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

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

A square matrix is a special type of matrix with an equal number of rows and columns. Learn more about square matrices in this lesson, including how to add and multiply them. Then, test your understanding with a short quiz.

What Is a Square Matrix?

A square matrix is a square array of numbers where the number of rows and columns are equal. The plural of matrix is matrices. Each number in the matrix is called an entry. Each entry is labeled based on its position in the matrix. For example, the entry in the first row and second column is labeled a with a subscript of 1, 2. You can label the entries of a matrix with any lower case letter, but you must keep the same letter within the same matrix.

These matrices are 3 x 3 matrices because they have 3 rows and 3 columns. A matrix can have entries of numbers, fractions, decimals, and even algebraic expressions. A square matrix can be any size as long as the numbers of rows and columns are equal. You can have a 2 x 2 square matrix as well as a 10 x 10 square matrix.

The Identity Matrix

While you can have any number of items in the entries, when the entries follow a special pattern, the matrix becomes a special type of square matrix. This is called the identity matrix because it has zeroes on all sides except on the diagonal, which are all ones. Identity matrices, just like regular square matrices, can be any size. They can be 2 x 2 or even 13 x 13. One interesting thing to note about the identity matrix is that when you multiply any matrix with its appropriately-sized identity matrix, it produces the matrix as the answer. The identity matrix doesn't change the matrix.

Adding Square Matrices

When it comes to adding square matrices, the only rule you have to follow is that the matrices you are adding have to be the same size. You can add 2 matrices that are both 3 x 3 or that are both 4 x 4. But if one matrix is 3 x 3 and the other is 5 x 5, then you can't add them.

If the two matrices are the same size, adding them is very straightforward.

What you do is you add each matching entry. You take the entry in the first matrix in the first row and first column position and add it to the entry in the second matrix that is also in the first row and first column position. You do this for each successive entry until you have added all entries together. If you were subtracting, you would do the same process as for addition, but you would be subtracting each entry instead of adding.

Multiplying Square Matrices

The rule of having two matrices be of the same size applies when you multiply them together. But when you multiply, you follow a slightly different procedure. It is a bit more complicated, but it's not difficult to remember. This is the formula for multiplying a 2 x 2 matrix:

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