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If A and B are two invertible square matrices of same order, then what is (AB)-1 equal to?

If A and B are two invertible square matrices of same order, then what is (AB)-1 equal to? Correct Answer B<sup>-1</sup>A<sup>-1</sup>

Concept:

If A is a square matrix;

(AB) - 1 = B - 1 A - 1

Calculation:

We know that AA - 1 = I

So, (AB) (AB) - 1 = I

Pre multiply by A - 1,

⇒ (A - 1 AB) (AB) - 1 = A - 1 I

⇒ (I B) (AB) - 1 = A - 1

⇒ B (AB) - 1 = A - 1

Pre multiply by B - 1,

⇒ B - 1 B (AB) - 1 = B - 1 A - 1

⇒ I (AB) - 1 = B - 1 A - 1

∴ (AB) - 1 = B - 1 A - 1

Note: (AB) - 1 ≠ A - 1 B - 1

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