Table of Contents

Toggle## Introduction

Every square matrix A is associated with a number, called its determinant and it is denoted by Δ or det (A) or |A|.

Note- Only square matrices have determinants.

## Types of Determinants and their Solving

### First order determinant

If A = [a], then det (A) = |A| = a

### Second order determinant

If A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} , then

det (A) = |A| = a_{11}a_{22} – a_{21}a_{12}

### Third order determinant

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

det (A) = |A| = a_{11}(a_{22}a_{33} – a_{32}a_{23}) – a_{12}(a_{21}a_{33} – a_{31}a_{23}) + a_{13}(a_{21}a_{32} – a_{22}a_{31})

## Evaluation of Determinant of Square Matrix of order 3 by Sarrus Rule

If A = \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}, then determinant can be formed by enlarging the matrix by adjoining the first two columns of the right and draw lines as shown below parallel and perpendicular to the diagonal.

Δ = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} – a_{13}a_{22}a_{31} – a_{11}a_{23}a_{32} – a_{12}a_{21}a_{33}.

*Note: This method doesn’t work for determinants of order greater than 3.*

**Properties of Determinants**

(i) The value of the determinant remains unchanged, if rows are changed into columns and columns are changed into rows.

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

(ii) The interchange of any two rows (or columns) of the determinant changes its sign.

{ \begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} } = -{ \begin{vmatrix} a_{21} & a_{22} & a_{23}\\ a_{11} & a_{12} & a_{13}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} }

(iii) If all the elements of a row (or column) are zero, then the determinant is zero.

\begin{vmatrix} 0 & a_{12} & a_{13}\\ 0 & a_{22} & a_{23}\\ 0 & a_{32} & a_{33} \end{vmatrix} = 0

(iv) If any two rows (or columns) of a determinant are identical, then its value is zero.

\begin{vmatrix} k & k & k\\ k & k & k\\ a_{31} & a_{32} & a_{33} \end{vmatrix} = 0

(v) If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.

k{ \begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} } = { \begin{vmatrix} ka_{11} & ka_{12} & ka_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} }

(vi) If each element of a row (or column) of a determinant is the sums of two or more terms, then the determinant can be expressed as the of two or more determinants

{ \begin{vmatrix} a_{11} + k & a_{12}+ k & a_{13}+ k\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} } = { \begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} +\begin{vmatrix} k & k & k\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} }

(vii) If the same multiple of the elements of any row (or column) of a determinant are added to the corresponding elements of any row (or column), then the value of the new determinant remains unchanged.

\begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} ={\begin{vmatrix} a_{11}+ka_{21} & a_{12}+ka_{22} & a_{13}+ka_{23}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix}}

## Minors and Cofactors

Minor of an element a_{ij} of a determinant is the determinant obtained by deleting its i^{th} row and j^{th} column in which element a_{ij} lies. Minor of an element a_{ij} is denoted by M_{ij}.

**Remark:** The minor of an element of a determinant of order n(n ≥ 2) is a determinant of order n – 1.

The cofactor of an element a_{ij}, denoted by A_{ij} is defined by A_{ij} = (–1)^{i + j} M_{ij}, where M_{ij} is minor of a_{ij}.

## Area of a Triangle

In earlier classes, we have studied that the area of a triangle whose vertices are (x_{1} , y_{1} ), (x_{2} , y_{2} ) and (x_{3} , y_{3} ), is given by the expression

Area = ½ [x_{1} (y_{2} – y_{3} ) + x_{2} (y_{3} –y_{1} ) + x_{3} (y_{1} –y_{2} )].

Now this expression can be written in the form of a determinant as

½{\begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1 \end{vmatrix} }

*Remarks*

- Since area is a positive quantity, we always take the absolute value of the determinant in (1).
- If the area is given, use both positive and negative values of the determinant for calculation.
- The area of the triangle formed by three collinear points is zero.

## Adjoint and Inverse of a Matrix

**Adjoint of a matrix:** The adjoint of a square matrix A = [a_{ij}]_{ n × n} is defined as the transpose of the matrix [A_{ij}] _{n × n}, where A_{ij} is the cofactor of the element a_{ij}. Adjoint of the matrix A is denoted by adj A.

Let A = \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix}

Then adj A = Transpose of \begin{pmatrix} A_{11} & A_{12} & A_{13}\\ A_{21} & A_{22} & A_{23}\\ A_{31} & A_{32} & A_{33} \end{pmatrix}

**Properties of the adjoint of a matrix**

If A and B are two non-singular matrices of the same order n, then

- A (adj A) = (adj A) A = |A| I
- (adj A
^{T}) = (adj A)^{T} - adj (AB) = adj B × adj A
- adj (adj A) = |A|
^{n-2} - |adj A| = |A|
^{n-1}

**Note : **

- Adjoint of a diagonal matrix is a diagonal matrix.
- Adjoint of a triangular Matrix is a triangular matrix
- Adjoint of a symmetric matrix is a symmetric matric

**Inverse of a Matrix : **A^{-1} = {1\over |A|}adj\ A

**Properties of Inverse Matrix**

Let A and B be two square matrices of the same order n. Then,

- (A
^{-1})^{-1}= A - (AB)
^{-1}= B^{-1}A^{-1} - (A
^{T})^{-1}= (A^{-1})^{T} - |A
^{-1}| = |A|^{-1} - A.A
^{-1}= A^{-1}A = I

## Singular and Non-Singular Matrix

**Singular Matrix:** A square matrix A is said to be singular if |A| = 0

**Non-Singular Matrix:** A square matrix A is said to be singular if |A| ≠ 0

**Theorem:**If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.**Theorem:**The determinant of the product of matrices is equal to the product of their respective determinants, that is, AB = AB, where A and B are square matrices of the same order.**Theorem:**A square matrix A is invertible if and only if A is a nonsingular matrix.

**Remark**

*In general, if A is a square matrix of order n, then |adj(A)| = |A| ^{n – 1}*

## Applications of Determinants and Matrices

**Consistent system:**A system of equations is said to be consistent if its solution (one or more) exists.**Inconsistent system:**A system of equations is said to be inconsistent if its solution does not exist.

**Solution of a system of linear equations using the inverse of a matrix **

Let us express the system of linear equations as matrix equations and solve them using the inverse of the coefficient matrix.

Consider the system of equations

a_{1} x + b_{1} y + c_{1} z = d_{1}

a_{2} x + b_{2} y + c_{2} z = d2

a_{3} x + b_{3} y + c_{3} z = d_{3}

Let A = \begin{pmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{pmatrix}; X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} and B = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}

**Case I –** If A is a nonsingular matrix, then its inverse exists. Now

AX = B

or, A^{–1} (AX) = A^{–1} B (premultiplying by A^{–1})

or, (A^{–1}A) X = A^{–1} B (by the associative property)

or, IX = A^{–1} B [∵ A^{–1}A = I]

or, X = A^{–1} B

This matrix equation provides a unique solution for the given system of equations as the inverse of a matrix is unique. This method of solving system of equations is known as the Matrix Method.

**Case II – **If A is a singular matrix, then |A| = 0.

In this case, we calculate (adj A) B.

- If (adj A) B ≠ O, (O being zero matrices), then the solution does not exist and the system of equations is called inconsistent.
- If (adj A) B = O, then the system may be either consistent or inconsistent according to the system has either infinitely many solutions or no solution

**Solution of a system of linear equations using the Cramer’s Rule**

Consider the system of equations

a_{1} x + b_{1} y + c_{1} z = d_{1}

a_{2} x + b_{2} y + c_{2} z = d2

a_{3} x + b_{3} y + c_{3} z = d_{3}

Then the solution of the system of equations is

x = Δ_x \over Δ; y = Δ_y \over Δ; z = Δ_z \over Δ

Where, Δ = {\begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{vmatrix} }

Δ_{x = } {\begin{vmatrix} d_{1} & b_{1} & c_{1}\\ d_{2} & b_{2} & c_{2}\\ d_{3} & b_{3} & c_{3} \end{vmatrix} }

Δ_{y }= {\begin{vmatrix} a_{1} & d_{1} & c_{1}\\ a_{2} & d_{2} & c_{2}\\ a_{3} & d_{3} & c_{3} \end{vmatrix} }

Δ_{z} = {\begin{vmatrix} a_{1} & b_{1} & d_{1}\\ a_{2} & b_{2} & d_{2}\\ a_{3} & b_{3} & d_{3} \end{vmatrix} }

- If D ≠ 0, then the system of equation is consistent with the unique solution.
- If D = 0 and at least one of the determinants D
_{1}, D_{2}, D_{3}is non-zero, then the given system is inconsistent, i.e. having no solution. - If D = 0 and D
_{1}= D_{2}= D_{3}= 0, then the system is consistent, with infinitely many solutions. - If D ≠ 0 and D
_{1}= D_{2}= D_{3}= 0, then systemhas only trivial solution, (x = y = z = 0).