Square Matrix

Once you've studied the basics of matrices, you can start looking at some specific types of matrices. One of the first you'll cover is a square matrix, which may be defined as any matrix with the same number of rows and columns.

For example,

is a square matrix because it has 4 horizontal rows and 4 vertical columns. In contrast,

is not a square matrix because it has 3 horizontal rows but only 2 vertical columns.

In this article, we'll explore how square matrices are classified and what you can do with them. Let's get started!

Square matrix nomenclature

A square matrix is often called a square matrix of order n, where n is the number of rows and columns (or dimensions) it has. For instance, the example of a square matrix above could be called a square matrix of order 4 because it measures $(4 \times 4)$ . Square matrices of the same order can always be added and multiplied.

Furthermore, square matrices have a main diagonal represented by an imaginary line beginning at the upper left and extending diagonally to the bottom right. In our example above, the main diagonal includes the elements 1, 3, 6, and 6. The sum of all of the elements on a square matrix's main diagonal is called the trace and is often denoted as Tr. For instance, a problem asking for the trace of matrix A would ask for $T r (A)$ .

The imaginary line from the upper right toward the bottom left is called the antidiagonal or counterdiagonal.

Applications of a square matrix

We can use square matrices to represent simple linear transformations such as shearing and rotations. For example, the square matrix

represents a counterclockwise rotation of 90 degrees in two-dimensional space. Similarly, we can use

to represent a reflection through the x-axis and

to increase the scaling by 2 in all directions. Square matrices have a variety of other uses in linear algebra as well.

Square matrix practice problems

a. If matrix $A$ is:

A =

Is matrix $A =$ a square matrix?

A square matrix must have the same number of horizontal rows and vertical columns. Matrix $A =$ measures $5 \times 5$ , so it is a square matrix with an order of 5.

b. If matrix $B$ is:

B =

Is matrix $B$ a square matrix?

A square matrix must have the same number of horizontal rows and vertical columns. Matrix $B$ measures $2 \times 4$ and is therefore not a square matrix.

c. Which elements are included in the main diagonal of the following square matrix:

M =

To solve this problem, we draw an imaginary line from the top left to the bottom right, noting all of the elements we hit along the way. That will be 9, 11, 4, and 10.

d. What is the trace of the square matrix above?

The formula for the trace is the sum of all of the elements on the main diagonal. We listed all of them above, so the trace is simply the sum of $(9 + 11 + 4 + 10 = 34)$ .

e. What is the trace of the following square matrix:

S =

First, we need to determine which elements are in the main diagonal of the square matrix. Drawing an imaginary line from the top left to the bottom right, we get 1, 5, and 9. Now, we have to add all 3 of those numbers together to find the trace. $(1 + 5 + 9 = 15)$ .

f. Write an example of a square matrix with an order of 6.

E =

Topics related to the Square Matrix

Adjoint of a Matrix

Identity Matrix

Cramer's Rule

Flashcards covering the Square Matrix

Linear Algebra Flashcards

Numerical Methods Flashcards

Practice tests covering the Square Matrix

Linear Algebra Diagnostic Tests

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Square matrices are an essential topic in linear algebra. Not only are they essential for linear transformations, but they also provide an introduction to more advanced concepts in the study of matrices. If you're struggling to determine whether a given matrix is a square matrix or finding the main diagonal of an $n \times n$ square matrix, a tutor could help you develop a deeper understanding of everything you're studying in class. Contact the Educational Directors at Varsity Tutors to learn more.

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