Inverse Of A Matrix (Reciprocal Matrix) :
Defination : A square matrix A said to be invertible (non singular) if there exists a matrix B such that
AB = I = BA
B is called the inverse (reciprocal) of A and is denoted by A1. Thus
A1 = B Û AB = I = BA
Properties of Inverse :
Property - 1 A-1 = (adj A) / (|A|)
We have, A .(adj A ) =|A| In
A-1 A(adj A ) = A-1 In |A|
In (adj A )= A-1 |A| In
A-1 = (A(adj A) /|A| )
Remarks : Note that adj A exists if A is non singular.
Note : The necessary and sufficient condition for a square matrix A to be invertible is that . |A| ¹ 0
Theorem 1 Every invertible matrix posses a unique inverse.
Proof : Let A be an invertible matrix of order n Χ n. Let B and C be two inverse of A.
Then, AB = BA = In..............(i)
and AC = CA = In ..............(ii)
Now, AB = In
Þ C(AB) = C In [pre-multipying by C ]
Þ (CA) B = C In [by associativity]
Þ (CA)B = C In [ CA = from (ii)]
Þ B = C [ In B = B, C In = C]
Hence an invertible matrix possesses a unique inverse.
Theorem 2 If A is an invertible square matrix, then AT is also invertible and (AT)1 = (A1)T
Proof : Since A is invertible matrix. Therefore,
[A ] ¹ 0
Þ |AT|¹ = 0 is also invertible. [AT|=|A|]
Now, AA1 = In = A1 A
Þ (AA1)T = (In)T = (A1A)T
Þ (A1)T (AT) = In = AT (A1)T
Þ (AT)1 = (A1A)T
Theorem 3 If A a non-singular matrix, then prove that
|A1| = | A|1 i.e. | A1| = (1/|A|)
Proof: Since |A| ¹ 0 therefore A1 exists such that
AA1 = I = A1 A
Þ |AA1| = |I|
Þ |A| |A1| = 1
[ |AB| = |A| |B| and |I| = 1]
Þ |A1| = (1/|A|) [|A|¹ 0 ]
Theorem4 If A & B are invertible matrices of the same order, then (AB)1 = B1 A1
Proof : A & B are invertible.
Þ |A| ¹ 0 & |B| ¹ 0 Þ |A| |B| ¹ 0 Þ |AB| ¹ 0
Þ AB is invertible
Let C = B1 A1
(AB) C = (AB) (B1 A1)
= A (BB1) A1 (assoviatively)
= AI A1 = (AI ) A1 = AA1 = AA1 = I ........(1)
Similarly C (AB) = I .............(2)
(1) and (2) Þ (AB) C = C (AB) = I Þ (AB)1 = C = B1 A1
Note :
(i) If A is invertible, (a) (A1)1 =A
(b) (Ak)1 = (A1)k
(ii) A square matrix is said to be orthogonal if, A1 = AT
(iii) |A1| = |A|1 i.e. |A1| = (1/|A|)
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