Question

Can a matrix of 5x5 matrices be in the subspace of 5x5 matrices?


For a matrix of 5x5 matrices:

1. Why are invertible matrices not in the Subspace of 5x5 matrices?

Eg. Invertible matrix A + invertible matrix B = matrix C that may not be invertible. So matrix C can not be in the subspace?

$A.B$= matrix is not invertible as there will never be a zero matrix in A or B? So $A.B$ will never be in the subspace?

2. Why are singular matrices not in the Subspace of 5x5 matrices?

Singular matrix D + singular matrix E = singular matrix F that may not be singular. So matrix F can not be in the subsp

Answer 1

Down the problem into smaller dimension to see what is going on.
\[A= \left[\begin{matrix}1&0\\1&1\end{matrix}\right]\rightarrow A^{-1}=\left[\begin{matrix}1&0\\-1&1\end{matrix}\right]\]

\[B=\left[\begin{matrix}1&0\\1&-1\end{matrix}\right]\rightarrow B^{-1}=\left[\begin{matrix}1&0\\1&-1\end{matrix}\right]\]

Hence both A, B are invertible. But.
\[C=A+B=\left[\begin{matrix}2&0\\2&0\end{matrix}\right]\] is not invertible.

Answer 2

Thank you for explaining =)