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M invertible -> N invertible for I&A\\B&C

Source: VTRMC 2004 P1

July 27, 2021

matrixlinear algebra

Problem Statement

Let

I

denote the

2\times2

identity matrix

\begin{pmatrix}1&0\\0&1\end{pmatrix}

and let

M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}

where

A,B,C

are arbitrary

2\times2

matrices which entries in

\mathbb R

, the real numbers. Thus

M

and

N

are

4\times4

matrices with entries in

\mathbb R

. Is it true that

M

is invertible (i.e. there is a

4\times4

matrix

X

such that

MX=XM=I

) implies

N

is invertible? Justify your answer.

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