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Adjoint and Inverse of a Matrix - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

Properties of adjoint of Matrix - Part 1 is considered one the most difficult concept.
Adjoint of a Matrix is considered one of the most asked concept.
29 Questions around this concept.

Solve by difficulty

EASY

If then adj $\left ( 3A^{2} +12A\right )$ is equal to :

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Concepts Covered - 3

Adjoint of a Matrix

Adjoint of a Matrix

Adjoint of a matrix A is the transpose of cofactor matrix of the matrix A. Cofactor matrix of matrix A is a matrix which has same order as that of A and has element $C_{ij}$ in place of $a_{ij}$ .

E.g.,

$\\\mathrm{let} \;A = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix} \\\\\mathrm{and \; let \; cofactor\; of \; every\; element\; is } \\ \\$

$\begin{bmatrix} C_{11} & C_{12} & C_{13}\\ C_{21} & C_{22} & C_{23}\\ C_{31} & C_{32} & C_{33} \end{bmatrix} \\\\\mathrm{then \; Adjoint\; of \; A\; is} \\\\A' = \begin{bmatrix} C_{11} & C_{12} & C_{13}\\ C_{21} & C_{22} & C_{23}\\ C_{31} & C_{32} & C_{33} \end{bmatrix}' = \begin{bmatrix} C_{11} & C_{21} & C_{31}\\ C_{12} & C_{22} & C_{32}\\ C_{13} & C_{23} & C_{33} \end{bmatrix}$

Properties of adjoint of Matrix - Part 1

Properties of adjoint of matrix

1. If A is a square matrix of order n, then

$\mathrm{(Adj \;A)A = A(Adj\;A ) = |A|\mathbb{I}_n}$ , or product of a matrix and its adjoint is commutative.

Proof :

$\\\mathrm{Let,\;A=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{bmatrix},\;then\;\;adj \;A =\begin{bmatrix}A_{11}&A_{21}&A_{31}\\ A_{12}&A_{22}&A_{32}\\ A_{13}&A_{23}&A_{33}\end{bmatrix}}\\\\\mathrm{where,A_{ij}\;\;is\;co-factor\;of\;a_{ij}}$

Since the sum of the product of elements of a row (or a column) with corresponding cofactors is equal to |A| and otherwise zero,

we have

$\mathrm{A\;(adj\;A)=\begin{bmatrix}\left|A\right|&0&0\\ 0&\left|A\right|&0\\ 0&0&\left|A\right|\end{bmatrix}=|A|\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}=|A|I}$

If A is a singular matrix of order n, then

(Adj A) A = A(Adj A) = O (null matrix) (As |A| = 0)

2. If A be non-singular square matrix of order n, then |Adj A| = |A|^n-1

Proof:

$\\\mathrm{A(Adj\;A)=|A|{I_n}}\\\\\mathrm{Taking\;determinant\;on\;both\;side}\\\\\mathrm{\left | A(Adj\;A) \right |=\left | |A|{I_n} \right |}\\\\\mathrm{\left | A\right |\left |(Adj\;A) \right |=\left | A \right |^n}\\\\\mathrm{|(adj\;A)|=|A|^{n-1}}$

3. If A and B are square matrices of order n, then, adj (AB) = (adj B) (adj A)

4. If A is a square matrix of order n, then, (adj A)’ = adj A’

Properties of adjoint of Matrix - Part 2

Properties of adjoint of matrix

4. If A be a square non-singular matrix of order n, then adj (adj A) = |A|^n-2A

Proof:

$\\\mathrm{A(adj A) = |A|\mathbb{I}_n} \\\mathrm{replace \; A\; by \; adj \; A,\; then} \\\mathrm{(adj \;A)(adj \; (adj\; A)) = |adj A|\mathbb{I}_n} =\mathrm{|A|^{n-1}\mathbb{I}_n}\\\mathrm{Pre-multiplying\; both \; sides\; by \; matrix \; A, then} \\\mathrm{A(adj \;A)(adj \; (adj\; A)) = A\mathbb{I}_n|A|^{n-1}=A|A|^{n-1}} \\\mathrm{|A|\mathbb{I}_n(adj \; (adj\; A)) =A|A|^{n-1}} \\\mathrm{(adj \; (adj\; A)) =A|A|^{n-2}=|A|^{n-2}A}$

5. If A is non-singular square matrix, then, $\mathrm{|adj (adj A)|= |A|^{(n-1)^2}}$

Proof: from the previous property, we know that

$\\\mathrm{adj (adj A)= |A|^{(n-2)}A} \\\mathrm{Taking\; determinant\; on \; both\; sides,} \\\mathrm{|adj (adj A)|=||A|^{(n-2)}A|} \mathrm{=|A|^{n(n-2)}|A| = |A|^{(n-1)^2}}\\\left ( \text{using}\;\;|kA|=k^n|A| \right )$