Question

Suppose that A is nonsingular. Prove that its inverse A-1 is unique. [Hint: Show that if there exists two matrices, B and C, such that BA = I and AC −=I, then necessarily B =C.]

Answer 1

Answer with Step-by-step explanation:

Let A is non-singular

`\mid A\mid\\eq 0`

We have to prove that
`A^(-1)` is unique.

Suppose B and C are inverse of A such that

`BA=I` and AC=I

By using property
`AA^(-1)=A^(-1)A=I`

`C=C`

`(BA)C=B(AC)`

`I\cdot C=B\cdot I`

`C=B`

Hence, the inverse of A is unique.