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

$\mid A\mid\neq 0$

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.