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linear transformation [Solved!]

My question

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

4. Multiplication of Matrices

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

T:M22-&gt;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 ??

Re: linear transformation

Hi Hans

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

T:M_{2,2}->M_{2,2} be defined by T[(a,b), (c, d)] = [(2c, a+c), (b-2c, d)].

Your attempts will appear as:

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 ??

(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?

My hint: What happens if you multiply both sides on the right by A^-1?

X

Hi Hans

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

<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?

Re: linear transformation

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

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?

Re: linear transformation

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

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?

Re: linear transformation

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

X

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