Matrix problem

**flintstone** · January 3rd 2010, 11:16 PM

If $A=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$ , find $c,d$

so that $(c I + d A)^2 = A$

**abender** · January 4th 2010, 12:01 AM

Originally Posted by flintstone

If $A=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$ , find $c,d$

so that $(c I + d A)^2 = A$

I will assume that c and d are scalar constants, while I is the indentitiy matrix. Let's start out by doing some of the small arithmetic:

$cI = \begin{pmatrix}c & 0 \\ 0 & c\end{pmatrix}$

$dA = \begin{pmatrix}0 & d \\ -d & 0\end{pmatrix}$

$ cI + dA = \begin{pmatrix}c & d \\ -d & c\end{pmatrix} $

$ (cI + dA)^2 = \begin{pmatrix}c & d \\ -d & c\end{pmatrix} \begin{pmatrix}c & d \\ -d & c\end{pmatrix} = \begin{pmatrix}c^2-d^2 & 2cd \\ -2cd & c^2-d^2\end{pmatrix} $

Now you should be able to set your c's and d's so that the matrix looks all like original A again.

-Andy

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