Let and be any two invertible square matrices of the same order.

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Let

A

and

B

be any two invertible square matrices of the same order. (i) State the relationship between the inverse of the product

AB

and the individual inverses of

A

and

B

. Explain in one or two sentences why the order of the factors is reversed in this relationship. (ii) Using the definition of an inverse matrix, give a short algebraic justification of the relationship stated in part (i) without expanding any determinants. (iii) Verify the result obtained in part (i) for

A=\begin{pmatrix}1&2\\0&1\end{pmatrix},\qquad B=\begin{pmatrix}3&0\\5&2\end{pmatrix}

by direct computation of

A^{-1},\,B^{-1}

and

(AB)^{-1}

. Compare your answer for

(AB)^{-1}

with

B^{-1}A^{-1}

. Offer a brief comment on your observation.

Related theory

Matrix Operations
Matrices
Determinants
Adjoint and inverse
Invertible matrices

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