4. Adjoint and Inverse of a Matrix

Adjoint and Inverse of a Matrix

The adjoint of a matrix (also called the adjugate of a matrix) is defined as the transpose of the cofactor matrix of that particular matrix. For a matrix A, the adjoint is denoted as adj (A). On the other hand, the inverse of a matrix A is that matrix which when multiplied by the matrix A give an identity matrix.

Note.: For a square matrix of order 2, given by

Let the determinant of a square matrix A be     |A|

Then the transpose of the matrix of co-factors is called the adjoint of the matrix A and is written as adj( A )

The product of a matrix A and its adjoint is equal to unit matrix multiplied by the determinant A.

Let A be a square matrix, then (Adjoint A). A = A. (Adjoint A) = | A |. I

Note:

 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 product of their respective determinants, that is, |AB| = |A|| B| , where A and B are square matrices of the same order.

Theorem : If A is a square matrix of order n, then |adj(A)| = |A|^{n – 1}.

Properties of adjoint matrix:

Let A and B  be two non singular square matrix of same order n,

• Property 1:  A (adj(A)) = (adj(A)) A = |A|I, where I is the identitiy matrix of order n.
• Property 2: A square matrix A is invertible if and only if A is a non-singular matrix.
• Property 3: adj(A’)= (adj (A))’
• Property 4: adj(kA)= k^n-1adj (A) , k Î R
• Property 5: adj(A^m)= (adj (A))^m
• Property 6: adj(adj(A))= |A|^n-2A
• Property 7: |adj(A)|= |A|^n-1
• Property 8: |adj(adj(A))|=|A|^(n-1)2 

Example : If A^T = – A then the elements on the diagonal of the matrix are equal to

(a) 1 (b) -1 (c) 0 (d) none of these

Solution:

(c) A^T = -A; A is skew-symmetric matrix; diagonal elements of A are zeros.

so option (c) is the answer.

Example : If A and B are two skew-symmetric matrices of order n, then,

(a) AB is a skew-symmetric matrix (b) AB is a symmetric matrix

(c) AB is a symmetric matrix if A and B commute (d)None of these

Solution:

(c) We are given A’ = -A and B’ = -B;

Now, (AB)’ = B’A’ = (-B) (-A) = BA = AB, if A and B commute.

Example : Let A and B be two matrices such that AB’ + BA’ = 0. If A is skew symmetric ,then BA

(a) Symmetric (b) Skew symmetric (c) Invertible (d) None of these

Solution:

(c) we have, (BA)’ = A’B’ = -AB’ [ A is skew symmetric]; = BA’ = B(-A)

= -BA

BA is skew symmetric.

Example : Which of the following statements are false –

(a) If | A | = 0, then | adj A | = 0;

(b) Adjoint of a diagonal matrix of order 3 × 3 is a diagonal matrix;

(c) Product of two upper triangular matrices is an upper triangular matrix;

(d) adj (AB) = adj (A) adj (B);

Solution:

(d) We have, adj (AB) = adj (B) adj (A) and not adj (AB) = adj (A) adj (B)

Inverse of a Matrix:

If A and B are two square matrices of the same order, such that AB = BA = I (I = unit matrix),Then B is called the inverse of A,

 i.e. B = A^–1 and A is the inverse of B.

Condition for a square matrix A to possess an inverse is that the matrix A is non-singular, i.e., | A | ≠ 0. If A is a square matrix and B is its inverse then AB = I.

Taking determinant of both sides | AB | = | I | or | A | | B | = I.

From this relation it is clear that | A | ≠ 0, i.e. the matrix A is non-singular.

We know that,

Properties of Inverse matrix:

Let A and B  be two non singular square matrix of same order n,

• Property 1:  (A^-1)^-1 = A
• Property 2: (AB)^-1=B^-1A^-1
• Property 3: (A’)^-1= (A^-1)’
• Property 4: (A^m)^-1= (A^-1)^m
• Property 5: |A^-1|= |A|^-1

Example:

=-28+30+18

=20

Similarly we can also obtain the values of B^-1 and A^-1 Then by multiplying B^-1 and A^-1