Square Matrix

A matrix is defined as the arrangement of elements in the form of an array. A square matrix is a type of matrix where the number of rows is equal to the number of columns. It is an effective way of arranging elements of the matrix. Learn the definition, concept, examples and operations on a square matrix in the following sections.

Definition of A Square Matrix

A square matrix is a matrix that has an equal number of rows and columns. The n x n matrix is called a square matrix. It is possible to add, subtract and multiply any two square matrices. The multiplication of two square matrices is also a square matrix. So, the number of elements of a square matrix is always a perfect square number.

The important matrices related to a square matrix are as follows:

  • Identity Matrix: It is a square matrix that has 1 as the diagonal elements and the remaining elements as zeros.
  • Scalar Matrix: It is a square matrix where the diagonal elements are constant numbers and others are equal to zero.
  • Trace of a Matrix: The total of the diagonal elements of a matrix is the trace of a matrix.

Also, Check

Square Matrix Examples

Here we are giving some of the examples of a square matrix with a detailed explanation.

\( A =\left[
\begin{matrix}
11 & 22\cr
33 & 44\cr
\end{matrix}
\right]
\)

The above mentioned matrix is a square matrix of order 2 x 2. The number of rows = 2 = number of columns. So, it is called a square matrix of order 2

\( B =\left[
\begin{matrix}
1 & 2 & 3\cr
4 & 6 & 8\cr
3 & 5 & 7\cr
\end{matrix}
\right]
\)

The above matrix has an order 3 x 3. Since the number of rows and columns are equal, it is a square matrix of order 3. We can find the determinant of the square matrix.

\( C =\left[
\begin{matrix}
14 & 10 & 3 & 11\cr
15 & 1 & 5 & 15\cr
7 & 13 & 2 & 8\cr
6 & 4 & 12 & 16\cr
\end{matrix}
\right]
\)

It is a square matrix of orde 4 x 4.

Operations of Square Matrices

Mathematical operations such as addition, subtraction, multiplication can be performed across two square matrices. Here we are giving the process and examples for a better understanding.

Addition & Subtraction of Two Square Matrices:

Two square matrices can be added/subtracted in a simple way. Let us consider two square matrices.

\(\left[
\begin{matrix}
a1 & a2\cr
a3 & a4\cr
\end{matrix}
\right] \) ±\(\left[
\begin{matrix}
b1 & b2\cr
b3 & b4\cr
\end{matrix}
\right] \) = \(\left[
\begin{matrix}
a1 ± b1 & a2 ± b2\cr
a3 ± b3 & a4 ± b4\cr
\end{matrix}
\right] \)

Examples:

\(\left[
\begin{matrix}
11 & 22\cr
33 & 44\cr
\end{matrix}
\right]
\) + \(\left[
\begin{matrix}
6 & 8\cr
15 & 10\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
17 & 30\cr
48 & 54\cr
\end{matrix}
\right]
\)

\(\left[
\begin{matrix}
7 & 16\cr
12 & 18\cr
\end{matrix}
\right]
\) – \(\left[
\begin{matrix}
5 & 9\cr
3 & 6\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
2 & 7\cr
9 & 12\cr
\end{matrix}
\right]
\)

Multiplication of Square Matrix:

Multiplication of a constant with a square is simple.

5A = 5 x \(\left[
\begin{matrix}
2 & 5\cr
10 & 13\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
10 & 35\cr
45 & 60\cr
\end{matrix}
\right]
\)

The multiplication of two square matrices involves a sequence of steps. We have to multiply the first row of the first matrix with the first column of the second matrix. The following is the detailed process.

\(A = \left[
\begin{matrix}
1 & 3\cr
2 & 4\cr
\end{matrix}
\right]
\) and \(B =\left[
\begin{matrix}
12 & 15\cr
4 & 6\cr
\end{matrix}
\right]
\)

A x B = \(\left[
\begin{matrix}
(1×12 + 2×4) & (1×15 + 3×6)\cr
(2×12 + 4×4) & (2×15 + 4×6)\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
20 & 33\cr
40 & 54\cr
\end{matrix}
\right]
\)

Transpose of a Square Matrix

The transpose of a matrix is the matrix obtained by transposing the elements of rows into columns, and columns into rows. The order of the given matrix and its transpose is different for some matrices. If the order of a matrix is m x n, then its transpose order is n x m. But in the case of a square matrix, the order is the same.

Example:

\( A =\left[
\begin{matrix}
8 & 10 & 12\cr
14 & 15 & 16\cr
3 & 5 & 7\cr
\end{matrix}
\right]
\)

AT = \(\left[
\begin{matrix}
8 & 14 & 3\cr
10 & 15 & 5\cr
12 & 16 & 7\cr
\end{matrix}
\right]
\)

In a square matrix, if the given matrix and its transpose are equal, then it is called a symmetric matrix. If the transpose matrix is equal to the negative of the given matrix, then it is a skew-symmetric matrix.

Square Matrix Determinant

The determinant of a matrix is a numerical value or it is a summary value that represents the entire set of elements of the matrix. The determinant of a square matrix having the order 2 x 2 can be easily calculated using the below-given formula.

\( A =\left[
\begin{matrix}
a & b\cr
c & d\cr
\end{matrix}
\right]
\)

|A| = |ad – bc|

If |A| = 0, then it is called a singular matrix otherwise it is a non-singular matrix.

The Inverse of a Square Matrix

The inverse of a matrix is used to divide one matrix with another matrix. You have to calculate the determinant of a square matrix and its adjoint to find its inverse. The inverse of a matrix is obtained by dividing the adjoint matrix with the det of the square matrix.

\( A =\left[
\begin{matrix}
a & b\cr
c & d\cr
\end{matrix}
\right]
\)

A-1 = \(\frac { 1 }{ |ad – bc| } \) . \(\left[
\begin{matrix}
d & -b\cr
-c & a\cr
\end{matrix}
\right]
\) = \(\frac { 1 }{ |A| } \) . adj(A)

A square matrix is called an orthogonal matrix if A T = A-1

Important Properties of Square Matrix

The important properties of a square matrix are given here:

  • The number of rows is equal to the number of columns.
  • The sum of the elements of a square matrix is the trace of a matrix.
  • The order of the transpose and original matrices are the same.
  • We can perform different operations like addition, subtraction, multiplication and inverse on a square matrix.
  • The determinant also can be calculated easily.

Example Questions & Answers

Question 1:
Find the transpose of the square matrix \( A =\left[
\begin{matrix}
4 & 9\cr
11 & 35\cr
\end{matrix}
\right]
\)

Answer:
Given matrix is \( A =\left[
\begin{matrix}
4 & 9\cr
11 & 35\cr
\end{matrix}
\right]
\)
It’s transpose is \(\left[
\begin{matrix}
4 & 11\cr
9 & 35\cr
\end{matrix}
\right]
\)

Question 2:
Find the multiplication of two square matrices \( A =\left[
\begin{matrix}
1 & 8\cr
7 & 5\cr
\end{matrix}
\right]
\) and \( B =\left[
\begin{matrix}
5 & 4\cr
6 & 7\cr
\end{matrix}
\right]
\)

Answer:
The given matrices are \( A =\left[
\begin{matrix}
1 & 8\cr
7 & 5\cr
\end{matrix}
\right]
\) and \( B =\left[
\begin{matrix}
5 & 4\cr
6 & 7\cr
\end{matrix}
\right]
\)
A x B = \(\left[
\begin{matrix}
1×5 + 8×6 & 1×4 + 8×7\cr
7×5 + 5×6 & 7×4 + 5×7\cr
\end{matrix}
\right]
\) = \(\left[
\begin{matrix}
53 & 60\cr
65 & 63\cr
\end{matrix}
\right]
\)

Question 3:
Find the inverse of the given square matrix \(C = \left[
\begin{matrix}
5 & 6\cr
2 & 3\cr
\end{matrix}
\right]
\)

Answer:
Given matrix is \(C = \left[
\begin{matrix}
5 & 6\cr
2 & 3\cr
\end{matrix}
\right]
\)
|C| = |ad – bc| = |15 – 12| = 3
C-1 = \(\frac { 1 }{ 3 } \) . \( \left[
\begin{matrix}
3 & -6\cr
-2 & 5\cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
1 & -2\cr
-2/3 & 5/3\cr
\end{matrix}
\right]
\)

FAQ’s on Square Matrix

1. What is a square matrix give an example?

A square matrix is a m x m matrix, where first m is the number of rows and second m is the number of columns. The example is \( A =\left[
\begin{matrix}
3 & 4 & 9\cr
12 & 11 & 35\cr
12 & 11 & 35\cr
\end{matrix}
\right]
\)

2. How to find a square matrix?

A square matrix can be found by checking the number of rows and columns in it. If they are equal, then it is said to be a square matrix.

3. What is squaring a matrix?

Squaring a matrix means multiplying a matrix by its own. Before multiplying the matrices, you have to check whether the number of columns of the first matrix is equal to the number of rows of the second matrix or not.

4. What are the dimensions of a square matrix?

The dimension of a square matrix is the number of rows by the number of columns. Here, both numbers should be equal.

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