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CBSE Sample Paper 2022-23 · Section A · 1 mark

Q10: (AB−1)−1(AB^{-1})^{-1} for non-singular AA⁠, BB

Sample 2022-23Ch 3 · MatricesCh 4 · Determinants
The question

If A, B are non-singular square matrices of the same order, then (AB−1)−1=(AB^{-1})^{-1} =

(a) A−1BA^{-1}B

(b) A−1B−1A^{-1}B^{-1}

(c) BA−1BA^{-1}

(d) ABAB

Exam answer

Using (PQ)−1=Q−1P−1(PQ)^{-1} = Q^{-1}P^{-1}⁠:

Why?

This is the reversal law: the inverse of a product is the product of the inverses in the reverse order. It holds because

(PQ)(Q−1P−1)=P(QQ−1)P−1=PP−1=I\begin{aligned} &(PQ)(Q^{-1}P^{-1}) \\ &= P(QQ^{-1})P^{-1} \\ &= PP^{-1} = I \end{aligned}

Here P=AP = A and Q=B−1Q = B^{-1}⁠. Both are invertible, because A and B are non-singular.

(AB−1)−1=(B−1)−1A−1=BA−1\begin{aligned} (AB^{-1})^{-1} &= (B^{-1})^{-1}A^{-1} \\ &= BA^{-1} \end{aligned}
Why?

The inverse of an inverse is the matrix itself: (B−1)−1=B(B^{-1})^{-1} = B⁠.

Watch out: matrix products do not commute, so BA−1BA^{-1} and A−1BA^{-1}B (option (a)) are different in general. Keep the order the reversal law gives.

Answer: (c) BA−1BA^{-1}

Where this comes from

Chapter 3, Matrices: Inverse of a product of matrices.

Quick lesson · Worked NCERT solutions · Interactive lesson

Also uses Chapter 4, Determinants.

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