matrices

[mbbx3adr]

Let A be a 4x2 real matrix and B be a 2x4 real matrix such that

AB = 1 0 -1 0
0 1 0 -1
-1 0 1 0
0 -1 0 1

Find BA.

 

[happy]

Double post.

http://www.mathgoodies.com/forums/topic ... C_ID=29739

 

[pka]

Here are two possible values for A & B. They are not unique!
\(\displaystyle \L
A = \left[ \begin{array}{rr}
\\ 1 & 0 \\ 0 & 1 \\ -1 & 0 \\ 0 & -1 \end{array} \right]\)

\(\displaystyle \L
B= \left[ \begin{array}{rrrr}
\\ 1 & 0 & -1 & 0\\ 0 & 1 & 0 & -1 \end{array} \right]\)

 

[soroban]

Hello, mbbx3adr!

Let \(\displaystyle A\) be a 4x2 real matrix and \(\displaystyle B\) be a 2x4 real matrix such that

\(\displaystyle AB\: = \:\begin{pmatrix}1&0&-1& 0\\ 0&1&0&-1\\ -1&0&1&0\\ 0&-1&0&1\end{pmatrix}\)

Find \(\displaystyle BA\)

As pka pointed out, the matrices are of the form:

\(\displaystyle A\;=\;k\cdot\begin{pmatrix}1&0\\0&1\\-1&0\\0&-1\end{pmatrix}\;\;\;\;B\;=\;\frac{1}{k}\cdot\begin{pmatrix}1&0&-1&0\\0&1&0&-1\end{pmatrix}\;\;\) for any nonzero constant \(\displaystyle k\)


Therefore: \(\displaystyle \,BA\;=\;\frac{1}{k}\cdot\begin{pmatrix}1&0&-1&0\\0&1&0&-1\end{pmatrix}\,\cdot\,k\cdot\begin{pmatrix}1&0\\0&1\\-1&0\\0&-1\end{pmatrix} \;= \;\L\begin{pmatrix}2&0\\0&2\end{pmatrix}\)