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# Invertible Matrices

#### definition

If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A and it is denoted by A^(– 1). In that case A is said to be invertible.
For example, let
A = [(2,3),(1,2)] and B = [(2,-3),(-1,2)] be two matrices.

Now AB =[(2,3),(1,2)][(2,-3),(-1,2)]

=[(4-3,-6+6),(2-2,-3+4)]=[(1,0),(0,1)] = I

Also BA = [(1,0),(0,1)] = I .Thus B is the inverse of A , in order words B = A^(–1) and A is inverse of B, i.e., A = B^(–1).

#### theorem

(Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique.
Proof:  Let A = [a_(ij)] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C.
Since B is the inverse of A
AB = BA = I                  ... (1)
Since C is also the inverse of A
AC = CA = I                  ... (2)
Thus   B = BI = B (AC) = (BA) C = IC = C

#### theorem

If A and B are invertible matrices of the same order, then (AB)^(–1) = B^(–1) A^(-1).

Proof: From the definition of inverse of a matrix, we have
(AB) (AB)^(–1) = 1

or A^(–1) (AB) (AB)^(–1) = A^(–1)I
(Pre multiplying both sides by A^(–1))
or (A^(–1)A) B (AB)^(–1) = A^(–1)
(Since A^(–1) I = A^(–1))

or IB (AB)^(–1) = A^(–1)

or B (AB)^(–1) = A^(–1)

or B^(–1) B (AB)^(–1) = B^(–1) A^(–1)

or I (AB)^(–1) = B^(–1) A^(–1)

Hence (AB)^(–1) = B^(–1) A^(–1)

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Matrices part 34 (Invertible matrices) [00:02:11]
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