Matrix Multiplication - PLEASE HELP!!!!!!

[jessica098]

I need help with a matrix multiplication problem please!

I am given the following 2x2 matrix (A):

[2 -1
-4 2]

I am asked to let X=

[a b
c d]

and come up with entries for X that satisfy the equation AX=0.

I've decided to come up with equations to help solve the problem, but I need help from there. The equations I've come up with are as follows:

2a-c=0
-4a+2c=0
2b-d=0
-4b+2d=0

Can anyone help me solve this to come up with entries for my matrix X?? Thanks!

 

[Denis]

Ever heard of google?

http://people.hofstra.edu/Stefan_Waner/ ... es3_2.html

 

[fasteddie65]

Re: Matrix multiplication

A trivial solution would be the zero matrix: [ 0 0 ]
[ 0 0 ]

 

[soroban]

Re: Matrix multiplication

Hello, Jessica!

\(\displaystyle \begin{bmatrix}2 & \text{-}1 \\ \text{-}4 & 2 \end{bmatrix}\begin{bmatrix}a& b\\ c& d \end{bmatrix} \;=\;\begin{bmatrix}0&0 \\ 0&0\end{bmatrix}\)

\(\displaystyle \text{I've come up with: }\:\begin{array}{ccccccc}2a-c\:=\:0 & [1] & & 2b-d \:=\:0 & [3] \\ \text{-}4a-2c \:=\:0 & [2] & & \text{-}4b+2d\:=\:0 & [4] \end{array}\)

Note that equations [1] and [2] are the same equation.
The same is true for equations [3] and [4].

So we have two equations in four variables.
There will be an infinite number of solutions.

From [1], we have: .\(\displaystyle c \,=\,2a\)

From [2], we have: .\(\displaystyle d \,=\,2b\)


\(\displaystyle \text{So any matrix of the form: }\:\begin{bmatrix} a & b \\ 2a & 2b\end{bmatrix}\: \text{ is a solution.}\)