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Many-One onto Function

If a matrix A...

Question

If a matrix A has inverses B and C, then which one of the following is correct ?

A

B may not be equal to C

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B

B should be equal to C.

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C

B and C should be unit matrices

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D

None of the above

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Solution

The correct option is D B should be equal to C.
The matrix A cannot have two inverse. Suppose $B A = I$ and $A C = I$ then $B = C$ . According to this proof by parentheses:
$B (A C) = (B A) C$
This implies $B I = I C$
Therefore $B = C$
This shows that a left-inverse $B$ (multiplying from the left) and a right-inverse $C$ (multiplying $A$ from the right to give $A C = I$ ) must be the same matrix.

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Many-One onto Function

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