Author Topic: Matrix Questions (Read 1625 times) Tweet Share

GerrySly · « **on:** March 22, 2009, 01:12:54 pm »

a. For which real values of x does the matrix $\begin{bmatrix} x & 2 \\ -1 & x + 2\\ \end{bmatrix}$ have an inverse?

b. For what values k does the system of equations

$\begin{align*} kx + 2y &= 4\\ x + (k - 1)y &= 2\\ \end{align*}$

not have a unique solution?

Not sure how to approach a, but with b I know that I must create a matrix equation to solve, so I get this

$\begin{bmatrix} k & 2 \\ x & k - 1\\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} 4 \\ 2\\ \end{bmatrix}$

Then attempting to solve I get this

$\begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} \frac{4}{k + 1} \\ \frac{2}{k + 1} \\ \end{bmatrix}$

Now I am not sure where to go from there only that k cannot equal -1, but that gives no solution not a unique solution (I think I may have missed a part in the course heh had a few days off when we were doing matrices)

Flaming_Arrow · « **Reply #1 on:** March 22, 2009, 01:51:47 pm »

a. are you sure you copied down the question right?

b.
$\begin{align*} kx + 2y &= 4\\ x + (k - 1)y &= 2\\ \end{align*}$

arrange for y

$y=\frac{4-kx}{2}$

$y= \frac{2-x}{k-1}$

equate the gradients

$\frac{-k}{2}=\frac{-1}{k-1}$

$k(k-1)=2$

$k^2-k-2=0$

$(k-2)(k+1) =0$

nullfactor law yields

$k=2 \mbox{or} k=-1$

pHysiX · « **Reply #2 on:** March 22, 2009, 02:20:52 pm »

a) for an inverse to exist, it's determinant must not be zero. sorry i can't read the latex so i can't really figure it out. hope it helps =]

*is the matrix:
[x 2]
[-1 (x+2)]

a oops...made a silly mistake =]

TrueTears · « **Reply #3 on:** March 22, 2009, 02:26:04 pm »

a)

the determinant is $x(x+2) - (2)(-1) = x^2+2x+2$

for a matrix to have an inverse the determinant must be larger than 0

Completing the square yields:

$x^2 + 2x + 1 -1 +2 = (x+1)^2 +1$

As we can see, for any values of x the y value will always be larger than 0

therefore $x \in R$

GerrySly · « **Reply #4 on:** March 23, 2009, 03:55:04 pm »

Bah, just did the test today, no matrix questions what so ever on it. So disappointed, I worked so hard revising matrices heh

Mao · « **Reply #5 on:** March 23, 2009, 07:54:12 pm »

FYI, all the matrices stuff you need to know for MM CAS are here: http://notes.vcenotes.com/?step=download&action=feedback&id=57

there really isn't much to do at all.

You do NOT need to worry about consistency, or inverse of a matrix, or determinants, or anything like that.
[also, just reading that over again, I've made a few small errors here and there, though for the sake of MM none of them actually matter. I wrote that before I started learnt it properly at uni.. oops

]

Search the forums now!

We have moved!

Author Topic: Matrix Questions (Read 1625 times) Tweet Share

GerrySly

Matrix Questions

Flaming_Arrow

Re: Matrix Questions

pHysiX

Re: Matrix Questions

TrueTears

Re: Matrix Questions

GerrySly

Re: Matrix Questions

Mao

Re: Matrix Questions

Recent Posts

User:
Password: