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Square Matrix – Explanation & Examples
A square matrix is a special type of matrix in which the number of rows equals the number of columns. In aesthetics, it takes the shape of a square. Firstly, let’s check the formal definition of a square matrix.
A matrix that has equal number of rows and columns is known as a square matrix.
In this article, we are going to take a close look at what square matrices are, how to square a matrix, the properties of square matrix, and the determinant of a square matrix. Let’s start!
What is a Square Matrix?
A square matrix is a special type of matrix in which the number of rows is equal to the number of columns. If a square matrix has $ n $ rows and $ n $ columns, it is said to be of the order $ n $.
Aesthetically, as the name suggests, the matrix looks like a square. Square matrices can be of the order $ 1 $, $ 2 $, or any number, $ n $. Theoretically we can have a square matrix of the order $ 100 $! But practically, it is tough to work with square matrices of order $ 11 $ and above.
Below, we show some of the most common square matrices.
$ \begin{bmatrix} a \end{bmatrix} $
This is a $ 1 \times 1 $ matrix. It has $ 1 $ row and $ 1 $ column. This is the simplest square matrix.
Next,
$ \begin{bmatrix} s & t \\ u & { v } \end {bmatrix} $
This is a $ 2 \times 2 $ matrix. It has $ 2 $ rows and $ 2 $ columns. This matrix is of the order $ 2 $.
Lastly, we have:
$ \begin{bmatrix} 3 & 4 & 5 \\ { – 1 } & { -3 } & 4 \\ { – 2 } & { – 8 } & 1 \end {bmatrix} $
This is a $ 3 \times 3 $ matrix. It has $ 3 $ rows and $ 3 $ columns. This matrix is of the order $ 3 $.
Of course, we can have square matrices of order $ 4 $, $ 5 $, or more, but they are not encountered until higher order linear algebra. For the purpose of this article, we will stick to square matrices of the order $ 1 $, $ 2 $, and $ 3 $ only.
Identity Matrix
This is a special type of square matrix where the values of a square matrix, other than its diagonal, are zeros. Below we show identity matrices of order $ 1 $, $ 2 $, and $ 3 $.
Identity Matrix of Order $ 1 $
$ \begin{bmatrix} 1 \end{bmatrix} $
Identity Matrix of Order $ 2 $
$ \begin{bmatrix} 1 & { 0 } \\ { 0 } & 1 \end {bmatrix} $
Identity Matrix of Order $ 3 $
$ \begin{bmatrix} 1 & { 0 } & { 0 } \\ { 0 } & 1 & { 0 } \\ { 0 } & { 0 } & 1 \end {bmatrix} $
The identity matrix is the matrix equivalent of the number “1”.
How to Square a Matrix
Square Matrix is a type of matrix.
But when we are talking about squaring a matrix, we are actually doing an operation of multiplying a matrix by itself. So, how do we square a matrix?
If we were to square a Matrix $ A $, we would multiply Matrix $ A $ by itself. It will follow the process of matrix multiplication. We show the squaring of a $ 2 \times 2 $ matrix below.
We have Matrix $ A $:
$ A = \begin{bmatrix} 1 & 1 \\ 3 & { -2 } \end {bmatrix} $
Let’s square the Matrix $ A $.
$ A^{2} =\begin{bmatrix} 1 & 1 \\ 3 & { -2 } \end {bmatrix} \times \begin{bmatrix} 1 & 1 \\ 3 & { -2 } \end {bmatrix} $
$ A^{2} = \begin{bmatrix} (1*1 + 1*3) & ( 1*1 + 1*-2 ) \\ ( 3*1 + -2*3) & (3*1 + -2*-2) \end {bmatrix} $
$ A^{2} = \begin{bmatrix} 4 & { – 1 } \\ { – 3 } & {7} \end{bmatrix} $
If you want to know how we did the multiplication, please read the article matrix multiplication.
Square Matrix properties
There are several properties of square matrices but for the purpose of this article, we will look some of the properties that fall into the scope of this lesson.
Property 1:
A square matrix is called a symmetric matrix if the elements of the matrix are symmetric with respect to the principal diagonal.
Examples of $ 2 \times 2 $ and $ 3 \times 3 $ symmetric matrices are shown below:
$ \begin{bmatrix} 1 & 7 \\ 7 & 5 \end {bmatrix} $
$ \begin{bmatrix} 1 & 1 & 5 \\ 1 & 3 & 4 \\ 5 & 4 & 2 \end {bmatrix} $
Property 2:
A square matrix is called a skew-symmetric matrix if the elements of the matrix are negatively symmetric with respect to the principal diagonal.
Examples of $ 2 \times 2 $ and $ 3 \times 3 $ skew-symmetric matrices are shown below:
$ \begin{bmatrix} 1 & { – 1 } \\ 1 & 3 \end {bmatrix} $
$ \begin{bmatrix} 1 & 3 & { – 2 } \\ { – 3 } & 1 & { 0 } \\ 2 & { 0 } & 4 \end {bmatrix} $
Property 3:
A square matrix is said to be diagonal if the elements of the matrix, other than the principal diagonal elements, are zeros.
An example of a $ 3 \times 3 $ diagonal matrix is shown below.
$ \begin{bmatrix} 1 & { 0 } & { 0 } \\ { 0 } & 4 & { 0 } \\ { 0 } & { 0 } & { – 2 } \end {bmatrix} $
Property 4:
A square matrix $ B $ is called orthogonal if $ B B^{T} = I $, where $B^{T}$ is the matrix’s transpose and $I$ is the identity matrix.
Property 5:
If $ B $ is a square matrix of the order $ n $, then its trace, denoted by $tr(B)$, is the sum of the elements in its principal diagonal.
Determinant of a Square Matrix
The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix. Let’s see the determinant of a $ 2 \times 2 $ matrix.
Let us take Matrix $ A $, shown below:
$ A = \begin{bmatrix} a & b \\ c & d \end {bmatrix} $
Then, the determinant of this matrix is:
$ det(A) = ad – bc $
The determinant of a $ 3 \times 3 $ matrix is a bit complicated and is left for you to look at here.
Let’s check some examples to clarify our understanding of square matrices.
Example 1
Give an example of a square matrix with order $ 4 $.
Solution
We haven’t looked at a square matrix of order $ 4 $. But it’s quite simple to write one out. Simply write out a matrix with $ 4 $ rows and $ 4 $ columns with sixteen elements inside it. Shown below:
$ T = \begin{pmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end {pmatrix} $
Example 2
A square matrix $ B $ is shown below:
$ B = \begin{bmatrix} { 0 } & 4 \\ { – 4 } & 2 \end {bmatrix} $
What is the square of Matrix $ B $?
Solution
In order to square a matrix, we multiply it by itself. Shown below:
$ B^{2} =\begin{bmatrix} { 0 } & 4 \\ { – 4 } & 2 \end {bmatrix} \times \begin{bmatrix} { 0 } & 4 \\ { – 4 } & 2 \end {bmatrix} $
$ B^{2} = \begin{bmatrix} (0*0 + 4*{ – 4 }) & (0*4 + 4*2) \\ ({ – 4 }*0 + 2*{ – 4 }) & ({ – 4 }*4 + 2* 2) \end {bmatrix} $
$ B^{2} = \begin{bmatrix} { – 16 } & 8 \\ { – 8 } & { – 12 } \end{bmatrix} $
Example 3
Calculate the determinant of Matrix $ P $.
$ P = \begin{bmatrix} { – 9 } & { – 5 } \\ { 0 } & 4 \end {bmatrix} $
Solution
Matrix $ P $ is a square matrix of order $ 2 $. We calculate the determinant of Matrix $P$ by using the determinant formula of a $ 2 \times 2$ matrix.
$ det(P) = ({ – 9 })(4) – ({ 0 })({ – 5 }) $
$ = { – 36 }$
Practice Questions
- Write an example of each of the following:
- $ 4 \times 4 $ diagonal matrix
- $ 3 \times 3$ identity matrix
- $ 3 \times 3$ symmetric matrix
- $ 2 \times 2$ skew-symmetric matrix
- What is the square of Matrix $ C $ shown below:
$ C = \begin{bmatrix} 1 & { 0 } \\ { – 2 } & 8 \end {bmatrix} $ - What is the trace of the $ 3 \times 3 $ matrix shown below:
$ S = \begin{bmatrix} 1 & { – 1 } & 9 \\ { 0 } & { – 4 } & 1 \\ 11 & { – 3 } & 1 \end {bmatrix} $ - Calculate the determinant of the $ 2 \times 2 $ matrix shown below:
$ G = \begin{pmatrix} 3 & 6 \\ { – 2 } & 8 \end{pmatrix} $
Answers
- An example of each of the following is shown below:
- $ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & { – 9 } & 0 \\ 0 & 0 & 0 & 11 \end {pmatrix} $
- $ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {bmatrix} $
- $ \begin{bmatrix} 1 & 1 & 7 \\ 1 & 4 & 2 \\ 7 & 2 & 3 \end {bmatrix} $
- $ \begin{bmatrix} { 3 } & { 4 } \\ { -4 } & { – 3 } \end {bmatrix} $
- Let’s calculate the square of Matrix $C$ by multiplying C by itself. Shown below:
$ C^{2} =\begin{bmatrix} 1 & { 0 } \\ { – 2 } & 8 \end {bmatrix} \times \begin{bmatrix} 1 & { 0 } \\ { – 2 } & 8 \end {bmatrix} $
$ C^{2} = \begin{bmatrix} (1*1 + 0*{ – 2 }) & (1*0 + 0*8) \\ (1*{ – 2 } + { – 2 }*8) & ({ – 2 }*0 + 8* 8) \end {bmatrix} $
$ C^{2} = \begin{bmatrix} 1 & { 0 } \\ { – 18 } & { 64 } \end{bmatrix} $
- We know the trace is the sum of the elements of the principal diagonal of a square matrix. Thus, we compute the trace as shown below:
$ tr( S ) = 1 + ({ – 4 }) + 1 = { – 2 } $
The trace of Matrix $ S $ is 2.
Matrix $ G $ is a square matrix of order $ 2 $. We calculate the determinant of Matrix $ G $ by using the determinant formula of a $ 2 \times 2 $ matrix.
$ det (G) = ( 3 )( 8 ) – ({ – 2 })( 6 ) $
$ det (G) = 24 + 12 $
$ det (G) = 36 $