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Chapter 2: Q 66E (page 100)

Question: Consider two n 脳 n matrices A and Bsuch that the product AB is invertible. Show that the matrices A and Bare both invertible.

Hint:AB(AB)鈭�1=In and(AB)鈭�1AB=In . Use Theorem 2.4.8.

Short Answer

Expert verified

It is proved that and are invertible.

Step by step solution

01

Use associative laws for the given matrices.

Consider the product of two n 脳 n matrices Aand Bsuch that AB is invertible.

The objective is to show that the matricesA andB are both invertible.

The matrixAB is invertible then there exists a matrixM such thatMAB=In and ABM=In.

Since, multiplication of matrices is associative so,

M(AB)=In(MA)B=In

And

(AB)M=InA(BM)=In

02

Show that A and B are invertible.

If A and Bare two matrices such that BA = ln thenA and Bare both invertible.

So,(MA)B=In implies the matrixB is invertible andABM=In implies the matrixA is invertible.

Hence, it is proved thatA and Bare invertible.

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