Properties of adjoint
The following are some properties of adjoint of a square matrix which are stated as theorems
Theorem–I: A (adj. A) = (adj. A) .A = |A| In.
Proof :
Þ A. (adj. A) = | A | I
If | A | ¹ 0 then (A(adj .A)/|A|) = I unit matrix of the dame order as that of A
Theorem –2 Let A be non- singular matrix os order n. Then
| adj A| = | A|n–1
Proof : A (adj A) = |A| In
Þ | A | | adj A | = | A |n Þ | adj A | | A |n–1 [ | AB | = | A | | B | ]
Theorem– 3 : If A and B are non singular square matrix of the same order , then
adj AB = (adj B) (adj A)
Proof : Since A and B are non singular square matrices of the same order , therefore AB exsists such that
| AB | = | A | | B | ¹ 0
( | A | ¹ 0, | B | ¹ 0 ]
We know that, (AB) (adj AB) = | AB | In ..............(i)
Also, (AB) (adj B adj A)
= (A (B. adj B) adj A
[By associativity]
= (A | B | In) adj A
[ B adj B = | B | In]
= | B | (A adj A)
[ A In = A]
= | B | ( | A | In)
[ A adj A = | A | In]
= | A | | B | In
= | A B | In
[ | A B | = | A | | B | ]
Thus, (AB) (adj B adj A) = | A B | In ..............(ii)
From (i) and (ii), we get (AB) (adj B) = (AB) (adj B . adj A)
But | AB |¹ 0 , therefore (AB)–1 exists. By cancellation law, we have
adj (AB) = adj B . Adj A
Theorem– 4 : If A is an invertible square matrix, then
adj AT = (adj A)T
Proof : Since A is invertible matrix. Therefore,
| A | ¹ 0
Þ | AT |¹ 0 [ | A | = | AT | ]
Þ | AT | is invertible
Now, A adj A = | A | In
Þ (A adj A)T = | A | In
Þ (adj A)T (AT)= | A | In ..............(i)
Also we have
(adj A)T (AT)= | AT | In
Þ (adj A)T (AT)= | A | In ...............(ii)
[ | A | = | AT | ]
From (i) and (ii) , we get
(adj AT) (AT)= (adj A)T (AT)
Þ Adj AT = (adj A)T [By right cancellation law]
Theorem– 5 :If A is a are non singular square matrix , then
adj (adj A) = | A |n–2 A
Proof : We know that
B (adj B) = | B | In for every square matrix of order n.
Replacing B by adj A, we get
(adj A [adj (adj A] = | adj A | In
Þ (adj A [adj (adj A] = | A |n–1 In [|adj A | = | A |n–1 ]
Þ A | (adj A) (adj (adj A)] = A { | A |n–1 In}
[pre-multiplying both sides by A]
Þ | A | In (adj adj A) = | A |n–1 A [ = AIn = A and A adj A = | A | In]
Þ | A | (In (adj adj A) = | A |n–1 A
Þ | A | (adj adj A) = | A |n–1 A
Þ adj adj A = | A |n–2 A
Theorem–6 : If A is a symmetric matrix, then adj A is also a symmetric matrix.
Proof : Let A be a symmetric matrix. Then,
AT = A .............(i)
We know that
adj (AT) = (adj A)T
[see theorem 4]
adj (A) = (adj A)T [using (i)]
Þ adj A is a symmetric matrix.
|
|
