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INVERSE OF A MATRIX
If A is a square matrix of order n and if there exists another matrix B of the same order such that
AB = BA = I, then B is called the inverse matrix of A and it is represented by A-1, and A is called invertible.
Example:
Hence, B is the inverse of A. Or B = A-1 and A = B-1.
Example:
Hence, B is the inverse of A. Or B = A-1 and A = B-1.
PROPERTIES OF INVERSE OF A MATRIX:
1. If the inverse of a square matrix exists, then it is unique.
Proof:
Let A = [aij] be a square matrix of order n and A-1 is its inverse. And let B be another matrix which is also an inverse of A. Then,
A A-1 = A-1A = I and AB = BA= I
Multiplying both sides of AB = I by A-1, we get
A-1(AB) = A-1I = A-1
(A-1A)B = A-1
Hence IB = A-1
Hence B = A-1
Hence the inverse of a square matrix is unique.
2. If A and B are two non-singular matrices (it means |A| ≠ 0 and |B| ≠ 0) of the same order, then AB is also not singular and (AB)-1 = B-1A-1, that is the inverse of the product of two non-singular matrices A and B is equal to the product of the inverses A-1 and B-1 in the reverse order.
Proof:
According to the definition of inverse of a matrix,
(AB)(AB)-1 = I
Multiplying both sides by A-1, we get
A-1(AB)(AB)-1 = A-1.I
Or (A-1A).B(AB)-1 = A-1 (Since A-1.I = A-1)
Or I.B(AB)-1 = A-1
Or B(AB)-1 = A-1
Or B-1B(AB)-1 = B-1A-1 (By multiplying both sides by B-1)
Or I.(AB)-1 = B-1A-1
Hence (AB)-1 = B-1A-1 (proved)
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