Determinant of a Square Matrix

Determinant of a square matrix is the scalar value or number calculated using the square matrix. The determinant of square matrix X is represented as |X| or det(X). In this article we will explore the determinant of square matrix in detail along with the determinant definition, determinant representation and determinant formula.

We will also discuss how to find determinant and solve some examples related to the determinant of a square matrix. Let's start our learning on the topic "Determinant of a Square Matrix".

What is Square Matrix?

A square matrix is a type of matrix in mathematics where the number of rows is equal to the number of columns. This means that a square matrix has an equal number of elements along its horizontal and vertical dimensions.

The general form of a square matrix of order n is represented as follows:

\mathbf{A} = \begin{bmatrix} 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{bmatrix}

What is Determinant of a Square Matrix?

Determinant of a square matrix is defined as the number obtained by the addition of the product of a row or column element with its cofactors. Determinant can only be obtained for a square matrix.

The determinant maps a square matrix to a specific number f: P→Q and is defined as f(p) = q, where q is the determinant of the square matrix p.

Determinant Representation

The determinant representation of a square matrix P is given below.

|P| or Det(P)

Determinant Formula for 2×2 Square Matrix

The formula for determinant of 2×2 square matrix A = \begin{bmatrix} a & b\\ c & d\\ \end{bmatrix} is given by:

|A| = ad - bc

Determinant Formula for 3×3 Square Matrix

The formula for determinant of 3×3 Square Matrix B = \begin{bmatrix} a& b& c\\ d& e& f\\ g& h& i\\ \end{bmatrix} is given by:

|B| = a [(e × i)- (f × h)] - b [(d × i)- (g × f)] + c [(d × h) - (e × g)]

Read More,

• Deteminant of Matrix
• Determinant of 2×2 Matrix
• Determinant of 3×3 Matrix

How to Find Determinant for n × n Square Matrix

To find the determinant for n × n square matrix we follow following steps:

• First, select any row or column of the given matrix.
• Then, for each element a_ij in the selected row find its cofactor C_ij.
• The cofactor of a_ij is given by: C_ij = (-1)^{i + j} M_ij where M_ij is minor of a_ij
• The minor of a_ij is the determinant obtained by eliminating i^th row and j^th column in the matrix.
• After finding all the cofactors of selected row/column element multiply each element with its cofactor and add them.
• The resultant gives the determinant of the given n × n square matrix.

Solved Examples on Determinant of Square Matrix

Example 1: Find the determinant of matrix X = \begin{bmatrix} 1 & 6\\ 14& -4 \end{bmatrix}

Solution:

To find the determinant of the given matrix we use formula:

|Matrix| = ad - bc

⇒ |X| = (1 × -4) - (14 × 6)

⇒ |X| = -4 - 84

⇒ |X| = -88

So, the determinant of given matrix is -88.

Example 2: Determine the determinant of matrix P = \begin{bmatrix} 0 & -5 & 3\\ 2& -1& 9\\ -7& 4& 6 \end{bmatrix}

Solution:

To find the determinant of matrix P we use formula:

|P| = a [(e × i)- (f × h)] - b [(d × i)- (g × f)] + c [(d × h) - (e × g)]

⇒ |P| = 0 [(-1 × 9)- (9 × 4)] - (-5) [(2 × 6)- (9 × -7)] + 3 [(2 × 4) - (-1 × -7)]

⇒ |P| = 0 [-9 - 36] + 5 [12 + 63] + 3[8 - 7]

⇒ |P| = 0 + 5 × 75 + 3 × 1

⇒ |P| = 375 + 3

⇒ |P| = 378

Example 3: Find the determinant of 4×4 matrix A = \begin{bmatrix} 0 & 1 & 2&-1\\ 3& -1& 2&4\\ -4& 0& 1& 2\\ 0&7& 3&-5 \end{bmatrix}

Solution:

To find the determinant we first select one row or column and alternate add and subtract the selected row elements with its cofactor.

Here, we select first row i.e., (0, 1, 2, -1)

Now, we calculate cofactors of each element.

Cofactor of a₁₁ i.e., 0 = (-1)¹⁺¹ \begin{vmatrix} -1 & 2 & 4\\ 0& 1& 2\\ 7& 3& -5 \end{vmatrix} = 11

Cofactor of a₁₂ i.e., 1 = (-1)¹⁺² \begin{vmatrix} 3 & 2 & 4\\ -4& 1& 2\\ 0& 3& -5 \end{vmatrix} = -121

Cofactor of a₁₃ i.e., 2 = (-1)¹⁺³ \begin{vmatrix} 3 & -1 & 4\\ -4& 0& 2\\ 0& 7& -5 \end{vmatrix} = -134

Cofactor of a₁₄ i.e., -1 = (-1)¹⁺⁴ \begin{vmatrix} 3 & -1 & 2\\ -4& 0& 1\\ 0& 7& 3 \end{vmatrix} = -89

Now, we add product of elements and its cofactors.

|A| = 0 ×11 + 1 × -121 + 2 × (-134) + (-1) × (-89)

⇒ |A| = 0 + -121 - 268 + 89

⇒ |A| = -300

So, the determinant of given 4×4 square matrix is -300.

Practice Questions on Determinant of Square Matrix

Q1: Find the determinant of matrix X = \begin{bmatrix} 12 & 8\\ 14& 23 \end{bmatrix}

Q2: Determine the determinant of matrix P = \begin{bmatrix} 20 & 15 & 10\\ 7& -11& 19\\ -4& 13& 6 \end{bmatrix}

Q3: Find the determinant of 4×4 matrix A = \begin{bmatrix} 0 & 5 & 3&-1\\ 3& -12& 9&6\\ -4& 0& 16& 2\\ 0&6& 8&-7 \end{bmatrix}

Q4: What is the determinant of matrix Q = \begin{bmatrix} 0 & -5 \\ 3& -12\\ \end{bmatrix}