Proving properties of matrix

**arze** · March 30th 2010, 08:09 PM

Let A be the matrix $\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$ , where no one of a,b,c,d is zero. It is required to find a non-zero 2x2 matrix X such that AX+XA=0, where 0 is the zero 2x2 matrix. Prove that either
(a) a+d=0, in which case the general solution for X depends on two parameters, or
(b) ad-bc=0, in which case the general solution for X depends on one parameter.

I don't know where to begin other than naming X= $\left(\begin{array}{cc}\\x_{11}&x_{12}\\x_{21}&x_{ 22}\end{array}\right)$ and then find the matrices AX and XA.
Thanks

**tonio** · March 31st 2010, 01:40 AM

Originally Posted by arze

Let A be the matrix $\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$ , where no one of a,b,c,d is zero. It is required to find a non-zero 2x2 matrix X such that AX+XA=0, where 0 is the zero 2x2 matrix. Prove that either
(a) a+d=0, in which case the general solution for X depends on two parameters, or
(b) ad-bc=0, in which case the general solution for X depends on one parameter.

I don't know where to begin other than naming X= $\left(\begin{array}{cc}\\x_{11}&x_{12}\\x_{21}&x_{ 22}\end{array}\right)$ and then find the matrices AX and XA.
Thanks

Exactly, that's what you have to do...and then solve $AX+XA=0$ , where the zero in the right is, of course, the zero 2x2 matrix.

Tonio

**arze** · March 31st 2010, 01:58 AM

Originally Posted by tonio

Exactly, that's what you have to do...and then solve $AX+XA=0$ , where the zero in the right is, of course, the zero 2x2 matrix.

Tonio

Ok, i did that and this is what I got

$AX=\left(\begin{array}{cc}{ax_{11}+bx_{21}}&{ax_{1 2}+bx_{22}}\\{cx_{11}+dx_{21}}&{cx_{12}+dx_{22}}\e nd{array}\right)$
$XA=\left(\begin{array}{cc}{ax_{11}+cx_{12}}&{bx_{1 1}+dx_{12}}\\{ax_{21}+cx_{22}}&{bx_{21}+dx_{22}}\e nd{array}\right)$
equating it with the zero matrix I get four equations.
$2ax_{11}+x_{12}(b+c)=0$ __1
$b(x_{22}+x_{11})+x_{12}(a+d)=0$ __2
$x_{21}(a+d)+c(x_{11}+x_{22})=0$ __3
$2dx_{22}+cx_{12}+bx_{21}=0$ __4

Now what do I do next?
Thanks

**tonio** · March 31st 2010, 02:37 AM

[quote=arze;484206]Ok, i did that and this is what I got

$AX=\left(\begin{array}{cc}{ax_{11}+bx_{21}}&{ax_{1 2}+bx_{22}}\\{cx_{11}+dx_{21}}&{cx_{12}+dx_{22}}\e nd{array}\right)$
$XA=\left(\begin{array}{cc}{ax_{11}+cx_{12}}&{bx_{1 1}+dx_{12}}\\{ax_{21}+cx_{22}}&{bx_{21}+dx_{22}}\e nd{array}\right)$
equating it with the zero matrix I get four equations.
$2ax_{11}+x_{12}(b+c)=0$ __1
$b(x_{22}+x_{11})+x_{12}(a+d)=0$ __2
$x_{21}(a+d)+c(x_{11}+x_{22})=0$ __3
$2dx_{22}+cx_{12}+bx_{21}=0$ __4

Now what do I do next?

Solve the system of equations, what else?! But first fix the first one: it must be

$2ax_{11}+bx_{21}+cx_{12}=0$

Tonio

**arze** · March 31st 2010, 02:44 AM

Oops! my bad
I have tried but can't seem to get them.

**tonio** · March 31st 2010, 03:49 AM

Originally Posted by arze

Oops! my bad
I have tried but can't seem to get them.

You have to work harder and make a bigger effort. The matrix of coefficients of the system ( in the unknonws $x_{11},x_{12},x_{21},x_{22}$ ) is:

$\begin{pmatrix}2a&c&b&0\\b&a+d&0&b\\c&0&a+d&c\\0&c &b&2d\end{pmatrix}$ . Divide now the first row by $2a$ (why can you?) and substract from each of the 2nd, 3rd rows a corresponding multiple of the first one to obtain:

$\begin{pmatrix}1&c\slash 2a&b\slash 2a&0\\0&a+d-bc\slash 2a&-b^2\slash 2a&b\\0&-x^2\slash 2a&a+d-bc\slash 2a&c\\0&c&b&2d\end{pmatrix}$ . Interchange now the 2nd and 4th rows, divide the new 2nd row by $c$ (again, why can you) and ...etc. This is

just the Gauss-Jordan reduction method of a matrix to echelon form!:

$\begin{pmatrix}1&c\slash 2a&b\slash 2a&0\\0&1&b\slash c&2d\slash c\\0&0&-\frac{b}{c}(a+d)&\left(\frac{b}{a}-\frac{2d}{c}\right)(a+d)\\0&0&a+d&\frac{c}{a}(a+d) \end{pmatrix}$ ...or something like this (check this carefully: I didn't!) ...and etc.

Tonio

**arze** · March 31st 2010, 04:01 AM

Originally Posted by tonio

$\begin{pmatrix}2a&c&b&0\\b&a+d&0&b\\c&0&a+d&c\\0&c &b&2d\end{pmatrix}$ .

I don't understand this part. How did you get that?
Thanks

Thread: Proving properties of matrix

LinkBack

Thread Tools

Display

Proving properties of matrix

Similar Math Help Forum Discussions

Proving properties of bumb function

Proving one of the laplace transform properties

Proving one of the Gamma Function's properties ..

Proving matrix inverse properties

Proving Vector Dot Product Properties

Search Tags