Matrix Operations

seogstar15 · **Joined:** Mon, 23 Feb 2009 01:43:57 UTC **Posts:** 5

let a = [1 0]
0 2 , find a matrix B such that AB - BA = [0 1]
-1 0

helmut · **Posted:** Mon, 23 Feb 2009 06:18:46 UTC

Any matrix of the form $B=\left( \begin{array}{cc} a & -1 \\ -1 & d \end{array} \right)$
will work.

Soroban · **Posted:** Mon, 23 Feb 2009 14:19:38 UTC

Hello, seogstar15!

Helmut is absolutely correct ... I assume you've verfied it.
Now how do we find this elusive matrix?

Quote:

Let $A = \begin{bmatrix}1&0\\ 0& 2\end{bmatrix}$
Find a matrix

such that: . $AB - BA \:=\:\begin{bmatrix}0& 1 \\ \text{-}1& 0\end{bmatrix}$

Let $B \:=\:\begin{bmatrix}a&b\\c&d\end{bmatrix}$

We have: . $\begin{array}{ccccccc}AB &=& \begin{bmatrix}1&0\\0&2\end{bmatrix}\begin{bmatrix}a&b\\c&d\end{bmatrix} &=& \begin{bmatrix}a&b\\2c&2d\end{bmatrix} \\ \\[-3mm] BA &=&\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}1&0\\0&2\end{bmatrix} &=& \begin{bmatrix}a & 2b \\c & 2d\end{bmatrix} \end{array}$

Then the equation becomes: . $\begin{bmatrix}a&b\\2c&2d\end{bmatrix} - \begin{bmatrix}a&2b\\c&2d\end{bmatrix} \;=\;\begin{bmatrix}0&1\\\text{-}1&2\end{bmatrix}$

. . . . . . . . . . . and we have: . $\begin{bmatrix}0&\text{-}b \\ c&0\end{bmatrix} \;=\;\begin{bmatrix}0&1\\\text{-}1&0\end{bmatrix}$

Therefore: . $\begin{bmatrix}a \:=\:\text{any} & & b \:=\:-1 \\ c \:=\:-1 & & d \:=\:\text{any} \end{bmatrix}$

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Matrix Operations

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