MathDB

Problem 1

Part of 2004 VTRMC

Problems(1)

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

Source: VTRMC 2004 P1

7/27/2021
Let II denote the 2×22\times2 identity matrix (1001)\begin{pmatrix}1&0\\0&1\end{pmatrix} and let M=(IABC),N=(IBAC)M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}where A,B,CA,B,C are arbitrary 2×22\times2 matrices which entries in R\mathbb R, the real numbers. Thus MM and NN are 4×44\times4 matrices with entries in R\mathbb R. Is it true that MM is invertible (i.e. there is a 4×44\times4 matrix XX such that MX=XM=IMX=XM=I) implies NN is invertible? Justify your answer.
matrixlinear algebra