linear transformation [Solved!]

X

T:M22->M22 be defined by T([matrix row 1: a b, row 2: c d])= [matrix row 1: 2c a+c, row 2: b-2c d].
To find the eigenvalues of T 
(This question is quoted from exercise of section 8.5 of book "Elementary Linear Algebra Application Version" by Howard Anton and Chris Rorres)

Relevant page

<a href="/matrices-determinants/4-multiplying-matrices.php">4. Multiplication of Matrices</a>

What I've done so far

I tried to find the [T][row 1: a b, row 2: c d]=[row 1: 2c a+c, row 2: b-2c d] but can't find a 2*2 matrix fits [T];
I also tried to separate [row 1: 2c a+c, row 2: b-2c d] into [row 1: c a, row 2:b d] + c[row 1: 1 1, row 2: -2 0] and then ??

X

Hi Hans

You are encouraged to use the math entry system in this forum to make your math easier to read.

Your question becomes:
<blockquote>
`T:M{2,2}-&gt;M{2,2}` be defined by `T[(a,b), (c, d)] = [(2c, a+c), (b-2c, d)].`
</blockquote>
Your attempts will appear as:
<blockquote>
I tried to find the `[T][(a,b), (c, d)] = [(2c, a+c), (b-2c, d)]` but can't find a 2*2 matrix fits [T];

I also tried to separate `[(2c, a+c), (b-2c, d)]` into `[(c, a), (b, d)] + c[(1, 1), (-2, 0)]` and then ??
</blockquote>

(You can click on the "Show code" button to see how I created those matrices.)

Let's put `A = [(a,b), (c, d)]` and `B = [(2c, a+c), (b-2c, d)]`.

We are trying to find a `2xx2` matrix `T` such that `TA = B`, right?

<b>My hint:</b> What happens if you multiply both sides on the right by `A^-1`?

X

Dear Murray,

Much thank for your hints, I worked out the matrix [T] as

 `1/(ad-bc)[(2cd-ac-c^2, ac-2bc+a^2),(bd-3cd,2bc+ad-b^2)]`

however I can't make the answer match with the textbook in which the answer of the eigenvalues are 1, -2, -1. Also I tried to work out the eigenvalue from `lambda*I-T`, the roots of the characteristic equation are very complicated that I think I am going in the wrong way, also I wonder how comes to have 3 eigenvalues from 2nd order characteristic equation?

X

Hi Hans

Your answer for T is correct, assuming it's matrix multiplication that the question was aiming for. I agree with you that solving the characteristic equation would be horrible, and there shouldn't be 3 eigenvalues! Is there something wrong with the question, or some context that we are both missing?

X

It seems Hans has disappeared. I'll close this topic in the absence of any further information.