Notes On Adjoint and Inverse of a Matrix - CBSE Class 12 Maths
The inverse of a square matrix obtaining by using elementary row or column operations. Adjoint of a matrix Consider A = $\left[\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right]$ Let Aij = Cofactor of aij   $\left[\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right]$  $↔$  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{12}}& {\text{A}}_{\text{13}}\\ {\text{A}}_{\text{21}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{23}}\\ {\text{A}}_{\text{31}}& {\text{A}}_{\text{32}}& {\text{A}}_{\text{33}}\end{array}\right]$    Transpose of matrix  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{12}}& {\text{A}}_{\text{13}}\\ {\text{A}}_{\text{21}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{23}}\\ {\text{A}}_{\text{31}}& {\text{A}}_{\text{32}}& {\text{A}}_{\text{33}}\end{array}\right]$  =  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}& {\text{A}}_{\text{31}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{32}}\\ {\text{A}}_{\text{13}}& {\text{A}}_{\text{23}}& {\text{A}}_{\text{33}}\end{array}\right]$  According to the definition, adj A =  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}& {\text{A}}_{\text{31}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{32}}\\ {\text{A}}_{\text{13}}& {\text{A}}_{\text{23}}& {\text{A}}_{\text{33}}\end{array}\right]$  Adjoint of a matrix Let A = $\left[\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right]$ The cofactors of the elements of A are A11 = d, A12 = -c, A21 = -b, A22 = a A = $\left[\begin{array}{cc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}\end{array}\right]$ =  Theorem 1 If A is any given square matrix of order n, then: A(adj A) = (adj A) A = |A| I, where I is the identity matrix of order n. Singular matrix: A square matrix is said to be singular if its determinant is zero. Non-singular matrix: A square matrix is said to be non-singular if its determinant is not zero. Theorem 2 If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order. Claim: |AB| = |A||B| Theorem 3 The determinant of the product of matrices is equal to the product of their respective determinants, that is, |AB| = |A||B| , where A and B are square matrices of the same order. Theorem 4 If A is a square matrix of order n, then |(Adj A)| = |A|n-1. Inverse of a square matrix: A square matrix, A, is said to be an invertible matrix if there exists a square matrix, B, such that AB = BA = I. If a square matrix, A, is invertible, then it is a non-singular matrix. Theorem 5 If A is a non-singular matrix, then A is invertible and A-1 = $\frac{\text{Adj A}}{\text{Det A}}$   .

#### Summary

The inverse of a square matrix obtaining by using elementary row or column operations. Adjoint of a matrix Consider A = $\left[\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right]$ Let Aij = Cofactor of aij   $\left[\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right]$  $↔$  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{12}}& {\text{A}}_{\text{13}}\\ {\text{A}}_{\text{21}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{23}}\\ {\text{A}}_{\text{31}}& {\text{A}}_{\text{32}}& {\text{A}}_{\text{33}}\end{array}\right]$    Transpose of matrix  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{12}}& {\text{A}}_{\text{13}}\\ {\text{A}}_{\text{21}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{23}}\\ {\text{A}}_{\text{31}}& {\text{A}}_{\text{32}}& {\text{A}}_{\text{33}}\end{array}\right]$  =  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}& {\text{A}}_{\text{31}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{32}}\\ {\text{A}}_{\text{13}}& {\text{A}}_{\text{23}}& {\text{A}}_{\text{33}}\end{array}\right]$  According to the definition, adj A =  $\left[\begin{array}{ccc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}& {\text{A}}_{\text{31}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}& {\text{A}}_{\text{32}}\\ {\text{A}}_{\text{13}}& {\text{A}}_{\text{23}}& {\text{A}}_{\text{33}}\end{array}\right]$  Adjoint of a matrix Let A = $\left[\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right]$ The cofactors of the elements of A are A11 = d, A12 = -c, A21 = -b, A22 = a A = $\left[\begin{array}{cc}{\text{A}}_{\text{11}}& {\text{A}}_{\text{21}}\\ {\text{A}}_{\text{12}}& {\text{A}}_{\text{22}}\end{array}\right]$ =  Theorem 1 If A is any given square matrix of order n, then: A(adj A) = (adj A) A = |A| I, where I is the identity matrix of order n. Singular matrix: A square matrix is said to be singular if its determinant is zero. Non-singular matrix: A square matrix is said to be non-singular if its determinant is not zero. Theorem 2 If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order. Claim: |AB| = |A||B| Theorem 3 The determinant of the product of matrices is equal to the product of their respective determinants, that is, |AB| = |A||B| , where A and B are square matrices of the same order. Theorem 4 If A is a square matrix of order n, then |(Adj A)| = |A|n-1. Inverse of a square matrix: A square matrix, A, is said to be an invertible matrix if there exists a square matrix, B, such that AB = BA = I. If a square matrix, A, is invertible, then it is a non-singular matrix. Theorem 5 If A is a non-singular matrix, then A is invertible and A-1 = $\frac{\text{Adj A}}{\text{Det A}}$   .

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