Determinants and Matrices

Determinants and matrices, in linear algebra, are used to solve linear equations by applying Cramer’s rule to a set of non-homogeneous equations which are in linear form. Determinants are calculated for square matrices only. If the determinant of a matrix is zero, it is called a singular determinant and if it is one, then it is known as unimodular. For the system of equations to have a unique solution, the determinant of the matrix must be nonsingular, that is its value must be nonzero. In this article, let us discuss the definition of determinants and matrices, different matrices types, properties, with examples.

Table of contents:

• Matrices
• Types of Matrices
• Inverse Matrix
• Transpose of Matrix
• Determinant
  • Properties
  • Laplace formula
• Determinant of a Matrix
• Examples
• FAQs

Matrices Definition

Matrices are the ordered rectangular array of numbers, which are used to express linear equations. A matrix has rows and columns. we can also perform the mathematical operations on matrices such as addition, subtraction, multiplication of matrix. Suppose the number of rows is m and columns is n, then the matrix is represented as m × n matrix.

Also, read:

• Inverse Of 3 By 3 Matrix
• Identity Matrix

Types of Matrices

There are different types of matrices. Let’s see some of the examples of different types of matrices

• Zero Matrix: \(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0\\ 0& 0 & 0 \end{bmatrix}\)
• Identity Matrix: \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0& 0 & 1 \end{bmatrix}\)
• Symmetric Matrix : \(\begin{bmatrix} 2 & 3 & -1 \\ 3 & 0 & 6\\ -1 & 6 & 5 \end{bmatrix}\)
• Diagonal Matrix: \(\begin{bmatrix} 6 & 0 & 0 \\ 0 & 9 & 0\\ 0 & 0 & 2 \end{bmatrix}\)
• Upper Triangular Matrix: \(\begin{bmatrix} 6 & -1 & 5 \\ 0 & 4 & 2\\ 0 & 0 & 2 \end{bmatrix}\)
• Lower Triangular Matrix: \(\begin{bmatrix} 6 & 0 & 0 \\ 2 & 4 & 0\\ 8 & -1 & 2 \end{bmatrix}\)

Inverse of a Matrix

Inverse of a matrix is defined usually for square matrices. For every m × n square matrix, there exists an inverse matrix. If A is the square matrix then A^-1 is the inverse of matrix A and satisfies the property:

AA^-1 = A^-1A = I, where I is the Identity matrix.

Also, the determinant of the square matrix here should not be equal to zero.

Transpose of Matrix

The transpose of a matrix can be determined by rows for the columns. If A is a matrix, then the transpose of a matrix is represented by A^T.

For example, let us assume a 3×3 matrix, Say A, then the transpose of A, i.e. A^T is given by

\(A =\begin{bmatrix} 6 & 2 & 1 \\ 3 & 5 & 9\\ 7 & 4 & 8 \end{bmatrix}\)

\(A^{T} =\begin{bmatrix} 6 & 3 & 7 \\ 2 & 5 & 4\\ 1 & 9 & 8 \end{bmatrix}\)

In case, if the given square matrix is a symmetric matrix, then the matrix A should be equal to A^T.

It means that A = A^T.

Definition of Determinant

A determinant can be defined in many ways for a square matrix.

The first and most simple way is to formulate the determinant by taking into account the top row elements and the corresponding minors. Take the first element of the top row and multiply it by it’s minor, then subtract the product of the second element and its minor. Continue to alternately add and subtract the product of each element of the top row with its respective minor until all the elements of the top row have been considered.

For example let us consider a 4×4 matrix A.

Second Method to find the determinant:

The second way to define a determinant is to express in terms of the columns of the matrix by expressing an n x n matrix in terms of the column vectors.

Consider the column vectors of matrix A as A = [ a₁, a₂, a₃, …a_n] where any element a_j is a vector of size x.

Then the determinant of matrix A is defined such that

Det [ a₁ + a₂ …. ba_j+cv … a_x ] = b det (A) + c det [ a₁+ a₂ + … v … a_x ]

Det [ a₁ + a₂ …. a_j a_j+1… a_x ] = – det [ a₁+ a₂ + … a_j+1 a_j … a_x ]

Det (I) = 1

Where the scalars are denoted by b and c, a vector of size x is denoted by v, and the identity matrix of size x is denoted by I.

We can infer from these equations that the determinant is a linear function of the columns. Further, we observe that the sign of the determinant can be interchanged by interchanging the position of adjacent columns. The identity matrix of the respective unit scalar is mapped by the alternating multi-linear function of the columns. This function is the determinant of the matrix.

Check: Determinant Of A 3×3 Matrix

Properties of Determinant

• If I_n is the identity matrix of the order nxn, then det(I) = 1
• If the matrix M^T is the transpose of matrix M, then det (M^T) = det (M)
• If matrix M^-1 is the inverse of matrix M, then det (M^-1) = \(\frac{1}{det (M)}\) = det (M)^-1
• If two square matrices M and N have the same size, then det (MN) = det (M) det (N)
• If matrix M has a size axa and C is a constant, then det (CM) = C^a det (M)
• If X, Y, and Z are three positive semidefinite matrices of equal size, then the following holds true along with the corollary det (X+Y) ≥ det(X) + det (Y) for X,Y, Z ≥ 0 det (X+Y+Z) + det C ≥ det (X+Y) + det (Y+Z)
• In a triangular matrix, the determinant is equal to the product of the diagonal elements.
• The determinant of a matrix is zero if all the elements of the matrix are zero.
• Laplace’s Formula and the Adjugate Matrix

Apart from these properties of determinants, there are some other properties, such as

• Reflection Property
• All-zero property
• Proportionality property or Repetition Property
• Switching Property
• Sum Property
• Scalar multiple Property
• Factor Property
• Triangle Property
• Invariance Property
• The determinant of Cofactor matrix

Laplace Formula for Determinant

With Laplace’s formula, the determinant of a matrix can be expressed in terms of the minors of the matrix.

If matrix B_xy is the minor of matrix A obtained by removing x^th and y^th column and has a size of

( j-1 x  j-1), then the determinant of the matrix A is given by

det (A) = \(\sum_{y=1}^{j}(-1)^{x+y}a_{x,y}B_{x,y}\)

And \((-1)^{x+y}B_{x,y}\) is known as the cofactor.

The adjugate matrix is obtained by transposing the matrix containing the cofactors and is given by the equation,

(Adj (A))_x,y = (-1)^x+y B_x,y

Class Related Links:

Determinant of a Matrix

To solve the system of linear equations and to find the inverse of a matrix, the determinants play an important role. Now, let us discuss how to find the determinant of 2×2 matrix and 3×3 matrix. If A is a matrix, then the determinant of a matrix A is generally represented using det (A) or |A|.

Finding Determinants for 2×2 matrix:

Let us assume a 2×2 square matrix

\(A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}\), then

|A| = \(\begin{vmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{vmatrix}\)

\(|A| = a_{11}a_{22}-a_{21}a_{12}\)

Finding Determinants for 3×3 Matrix

Now, assume the 3×3 matrix, say

\(A = \begin{bmatrix} a_{11} &a_{12} & a_{13}\\ a_{21} & a_{22} &a_{23} \\ a_{31} & a_{32} &a_{33} \end{bmatrix}\), then

\(|A| = \begin{vmatrix} a_{11} &a_{12} & a_{13}\\ a_{21} & a_{22} &a_{23} \\ a_{31} & a_{32} &a_{33} \end{vmatrix}\)

\(|A| = a_{11}\begin{bmatrix} a_{22} & a_{23}\\ a_{32} & a_{33} \end{bmatrix}- a_{12}\begin{bmatrix} a_{21} & a_{23}\\ a_{31} & a_{33} \end{bmatrix} + a_{13}\begin{bmatrix} a_{21} & a_{22}\\ a_{31} & a_{32} \end{bmatrix}\)

Determinants and Matrices Examples

Example 1: 

Find the determinant of matrix \(A = \begin{bmatrix} 4 & 2\\ 3& 2 \end{bmatrix}\)

Solution:

Given: \(A = \begin{bmatrix} 4 & 2\\ 3& 2 \end{bmatrix}\)

The determinant of matrix A is

det (A) = |A| = 8 – 6

|A| = 2

Example 2: 

Find the determinant of a matrix \(A = \begin{bmatrix} 2 & 3 & 1\\ 6 & 5 & 2 \\ 1 & 4 & 7 \end{bmatrix}\)

Solution:

Given: \(A = \begin{bmatrix} 2 & 3 & 1\\ 6 & 5 & 2 \\ 1 & 4 & 7 \end{bmatrix}\)

Therefore, 

\(|A| = 2\begin{vmatrix} 5 & 2\\ 4 & 7 \end{vmatrix} – 3\begin{vmatrix} 6 & 2\\ 1 & 7 \end{vmatrix} + 1\begin{vmatrix} 6 & 5\\ 1 & 4 \end{vmatrix}\)

|A| = 2(35-8) – 3(42-2) +1(24-5)

|A| = 2(27) – 3(40) + 1(19)

|A| = 54-120+19

|A| = 73 -120

|A| = -47

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