ISC (Commerce) Class 12CISCE
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# Adjoint and Inverse of a Matrix

#### definition

Defination:  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 = [(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)]

Then adj A = Transpose of [(A_11,A_12,A_13),(A_21,A_22,A_23),(A_31,A_32,A_33)] = [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]

#### theorem

If A be any given square matrix of order n, then
where I is the identity matrix of order n
Verification
Let A =[(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)] , then adj A = [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]
Since sum of product of elements of a row (or a column) with corresponding cofactors is equal to |A| and otherwise zero, we have
A (adj A) =  |(|A|,0,0),(0,|A|,0),(0,0,|A|)| = |A| [(1,0,0),(0,1,0),(0,0,1)]
Similarly, we can show  (adj A) A = A I

#### notes

Definition :  A square matrix A is said to be singular if A = 0.
For example, the determinant of matrix A = [(1,2),(4,8)] is zero
Hence A is a singular matrix.

Definition :  A square matrix A is said to be non-singular if A ≠ 0
Let A = [(1,4),(3,2)]. Then |A| = |(1,4)(3,2)| = 4 - 6 = 2 ≠ 0
Hence A is a nonsingular matrix.
We state the following theorems without proof.

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
Remark:  We know that (adj A) A = |A| I = [(|A|,0,0),(0,|A|,0),(0,0,|A|)] , |A| ≠ 0
Writing determinants of matrices on both sides, we have
|(adj A)A| = |(|A|,0,0),(0,|A|,0),(0,0,|A|)|

i.e,. |(adjA)| |A| = |A|^3 |(1,0,0),(0,1,0),(0,0,1)|

i.e. |(adj A)| |A| = |A|3 (1) i.e. |(adj A)| = |A|2

i.e. |(adj A)| = |A|^2

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

Theorem:  A square matrix A is invertible if and only if A is nonsingular matrix.

Proof:  Let A be invertible matrix of order n and I be the identity matrix of order n. Then, there exists a square matrix B of order n such that
AB = BA = I
Now   AB = I.  So |AB| = I or |A| |B| = 1  (since |I| =1, |AB|=|A||B|)
This gives |A| ≠ 0.
Hence A is nonsingular.
Conversely, let A be nonsingular. Then A ≠ 0
Now A (adj A) = (adj A) A = |A| I                  (Theorem 1)
or A(1/|A| adj A) = (1/|A| adj A) A = I
or AB = BA = I ,where  B = 1/|A| adj A
Thus, A is invertible and A^-1 = 1/ |A|adj A

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Determinants Part 4- Adjoint and Inverse of a Matrix [00:25:43]
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