Square Matrix
A square matrix is an important format of a matrix and it has the perfect square number of elements. It has an equal number of rows and columns, and hence its order is of the form n × n. All the matrix operations of transpose, determinant, adjoint, inverse, and the mathematical operations of matrices apply to a square matrix also.
A square matrix has special application in solving quadratic equations in two variables. Here we shall learn the different properties of a square matrix, and try to understand how to do the mathematical operations across these matrices.
What Is a Square Matrix?
A square matrix is a matrix with an equal number of rows and columns. Its order is of the form n x n. Also, the product of these rows and columns gives the number of elements in the square matrix. Hence the number of elements in it is always a perfect square number. A typical square matrix looks as follows.
\(A_{n,n} = \begin{pmatrix}a_{11} & a_{12} & \cdots & a_{1n} \\a_{21} & a_{22} & \cdots & a_{2n} \\\vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}\)
Here are some examples of square matrices.
 \(\left(\begin{array}{ll}
1 & 2 \\ \\
3 & 4
\end{array}\right)\) is a square matrix of order 2x2 (or simply, order 2).
It has 2x2 = 4 elements.  \(\left(\begin{array}{ccc}
3 & 6 & 2 \\
2 & 3 & 6 \\
6 & 2 & 3
\end{array}\right)\) is a square matrix of order 3x3 (or simply, order 3).
It has 3x3 = 9 elements.
Transpose of a Square Matrix
A matrix obtained by transforming the rows into columns and the columns into rows is called the transpose of the given matrix. Generally, the order of the matrix is changed on changing it into a transpose. For a matrix having an order m × n, the transpose of the matrix has an order of n × m. Here for a square matrix, the transpose matrix has the same order as the given matrix.
A = \(\begin{pmatrix}a & b & c\\d&e&f\\g&h&i\end{pmatrix}\) A^{T} = \(\begin{pmatrix}a & d & g\\b&e&h\\c&f&i\end{pmatrix}\)
Let us try to understand two important terms related to the transpose of a matrix. A square matrix whose transpose is equal to the given matrix is called a symmetric matrix. A square matrix whose transpose is equal to the negative of the given matrix is called a skewsymmetric matrix.
Symmetric Matrix: A^{T} = A
SkewSymmetric Matrix: A^{T} = A
Determinant of a Square Matrix
The determinant of a square matrix is a single numeric value or is a summary value representing the entire set of elements of the matrix. The determinant for a matrix of order 2 × 2 can be easily calculated using the formula. The determinant of a matrix of order 2 × 2 is equal to the difference of the product of the diagonal elements of the matrix. This can be observed in the below working.
A = \(\begin{pmatrix}a &b\\\\c&d\end{pmatrix}\)
A = ad  bc
Additionally, let us check a few matrices which have been termed based on their determinant values. The matrix whose determinant is equal to zero is called a singular matrix (A = 0). For a singular matrix, the inverse of a matrix does not exist. And the matrix whose determinant value is a nonzero value is called a nonsingular matrix (A ≠ 0).
The Inverse of a Square Matrix
The inverse of a square matrix is used to divide one matrix with another matrix. For finding the inverse of a matrix we need to first find the determinant of the matrix. Then the adjoint of the square matrix is calculated. Finally, the inverse is obtained by dividing the adjoint matrix with the determinant of the square matrix.
A = \(\begin{pmatrix}a &b\\\\c&d\end{pmatrix}\) A^{1} = \(\dfrac{1}{A}.Adj A\) = \(\frac{1}{ad  bc}\begin{pmatrix}d &b\\\\c&a\end{pmatrix}\)
Let us try to look into an important term relating to the inverse of a matrix. A square matrix is called an orthogonal matrix if its ranspose is equal to its inverse.
Orthogonal Matrix: A^{T} = A^{1}
Matrix Operations of a Square Matrix
The mathematical operations of addition, subtraction, and multiplication can also be performed across two square matrices. For addition or subtraction, the corresponding elements are added to obtain the resultant matrix. Matrix addition follows commutative property (A + B = B + A).
A + B= \(\begin{pmatrix}a &b\\\\c&d\end{pmatrix}\) + \(\begin{pmatrix}1 &2\\\\3&4\end{pmatrix}\) = \(\begin{pmatrix}a \pm1 &b \pm2\\\\c \pm3&d \pm4\end{pmatrix}\)
Matrix scalar multiplication is very simple. For the multiplication of a square matrix with a constant, each element of the matrix is multiplied with the constant.
kA= k\(\begin{pmatrix}a &b \\\\c &d \end{pmatrix}\) = \(\begin{pmatrix}ka &kb \\\\kc &kd \end{pmatrix}\)
The multiplication of two square matrices involves a sequence of steps. Let us consider two square matrices of order 2 × 2. Here we consider the row elements in the first matrix and the column elements in the second matrix. The following calculations show the sequence of calculations involved in the process of matrix multiplication.
Important Terms Relating to Square Matrix
Let us check a few important terms relating to the square matrix, which will help in our deeper understanding of the concepts of a square matrix.
 Order of a Matrix: It is the product of rows and columns in a matrix. A square matrix has an equal number of rows and columns and its order is n × n.
 Trace of a Matrix: It is equal to the sum of the diagonal elements of a square matrix.
 Identity Matrix: It is a square matrix and has ones as its diagonal elements, and all the other elements are zeros.
 Scalar Matrix: A square matrix having the same number as all its diagonal elements and all the other elements are equal to zero.
 Symmetric Matrix: A matrix whose transpose is equal to the given matrix is called a symmetric matrix.
 SkewSymmetric Matrix: A matrix whose transpose is equal to the negative of the same matrix is called a skewsymmetric matrix.
 Orthogonal Matrix: A matrix is said to be an orthogonal matrix if the inverse of the matrix is equal to the transpose of the matrix.
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Examples on Square Matrix

Example 1: Find the transpose of the matrix A = \(\begin{pmatrix} 4 & 2 \\\\5 & 7\end{pmatrix}\)
Solution:
The transpose of the given square matrix is obtained by transforming the row elements into column elements.
Given A = \(\begin{pmatrix} 4 & 2 \\\\5 & 7\end{pmatrix}\)
A^{T} = \(\begin{pmatrix} 4 & 5 \\\\2 & 7\end{pmatrix}\)

Example 2: Find the inverse of the square matrix A = \(\begin{pmatrix} 3 & 1 \\4 & 5\end{pmatrix}\).
Solution:
The given matrix A = \(\begin{pmatrix} 3 & 1 \\\\4 & 5\end{pmatrix}\)
As per formula if A = \(\begin{pmatrix}a &b\\\\c&d\end{pmatrix}\) then A^{1} = \(\frac{1}{ad  bc}\begin{pmatrix}d &b\\\\c&a\end{pmatrix}\)
Therefore A^{1} = \(\frac{1}{(3×(5)  1 × 4)}\begin{pmatrix} 5 & 1 \\\\4 & 3\end{pmatrix}\)
= \(\frac{1}{19}\begin{pmatrix} 5 & 1 \\\\4 & 3\end{pmatrix}\)

Example 3: Find the product of two square matrices A = \(\begin{pmatrix} 4 & 8 \\\\2 & 7\end{pmatrix}\) and B = \(\begin{pmatrix} 3 & 2 \\\\0 & 5\end{pmatrix}\).
Solution:
The given two matrices are A = \(\begin{pmatrix} 4 & 8 \\\\2 & 7\end{pmatrix}\) and B = \(\begin{pmatrix} 3 & 2 \\\\0 & 5\end{pmatrix}\)
A × B = \(\begin{pmatrix} 4×3 + 8×0 & 4×(2) + 8×5 \\\\2×3 + 7×0 & 2×(2) + 7 × 5\end{pmatrix}\)
= \(\begin{pmatrix} 12 & 32 \\\\6 & 31\end{pmatrix}\)
FAQs on Square Matrix
What is the Definition of a Square Matrix?
A square matrix is a matrix in which the number of rows = the number of columns. For example, matrices of orders 2x2, 3x3, 4x4, etc are square matrices. Matrices of orders like 2x3, 3x2, 4x5, etc are NOT square matrices (these are rectangular matrices).
How to Find the Inverse of a Square Matrix?
The inverse of a square matrix is obtained after finding the determinant and the adjoint of the given matrix. For a matrix A = \(\begin{pmatrix}a &b\\\\c&d\end{pmatrix}\) its determinant is A = ad  bc and Adj A =\( \begin{pmatrix}d &b\\\\c&a\end{pmatrix}\). Combining these two concepts the inverse of the matrix A is A^{1} = \(\dfrac{1}{A}.Adj A\) = \(\frac{1}{ad  bc}\begin{pmatrix}d &b\\\\c&a\end{pmatrix}\). The inverse of a matrix is only possible for nonsingular matrixes. A non singular matrix is a matrix whose determinant is a non zero value A ≠ 0.
How to Find If a Square Matrix Is Invertible?
A square matrix is invertible only if the determinant of the matrix is a non zero value A ≠ 0. For a matrix A = \(\begin{pmatrix}a &b\\\\c&d\end{pmatrix}\) its determinant is A = ad  bc. For matrices having A = 0, its inverse does not exist.
How to Multiply Square Matrices?
Two square matrices can be multipled if both the matrices are of the same order. The rows of the first matrix are considered and the columns of the second matrix are considered for multiplication.
How to Find the Determinant of a Square Matrix?
The determinant of a square matrix of order 2 × 2 is equal difference of the product of the elements of the first diagonal and the second diagonal. For a matrix A = \(\begin{pmatrix}a &b\\ \\c&d\end{pmatrix}\). it determinant is A = ad  bc.
What Is Magic Square Matrix?
A magic square matrix is a square matrix having an odd number of rows and columns (n), such that the sum of the elements in each row, or column, or a diagonal is equal. This sum of the elements is equal to n(n^{2} + 1)/2.
What Is the Order of a Square Matrix?
The order of a square matrix is of the form n × n, and it has an equal number of rows and columns. The number of elements in the square matrix can be calculated from its order and is equal to the square number n^{2}.
What Is the Transpose of a Square Matrix?
The transpose of a square matrix is obtained by writing the row elements of the given matrix as column elements. For a matrix A = \(\begin{pmatrix}a & b & c\\c&e&f\\g&h&i\end{pmatrix}\) its transpose is A^{T} = \(\begin{pmatrix}a & d & g\\b&e&h\\c&f&i\end{pmatrix}\). The elements of the first row of the given matrix has been written in the first column, and the elements of the second row of the matrix has been written in the second column, to obtain the transpose matrix.
What Is NonSingular Square Matrix?
A nonsingular matrix has a non zero value for its determinant(A ≠ 0. For a matrix A =\(\begin{pmatrix}a &b\\\\c&d\end{pmatrix}\), the condition for it to be a non singular matrix is ad  bc ≠ 0. The inverse of a given square matrix exists only if it is a non singular matrix.
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