T invariant subspace (intro lin alg class undergrad)

[david118]

Homework Statement

V=Matrix (2x2), T(A) = (0 1 ) A , and W = {A$\epsilon$ V: A$^{}t$ = A
(1 0)

Homework Equations

So T(A) transformation, multiplies a 2x2 matix with entries 0 1 1 0 by A with A on the right side

The Attempt at a Solution

I said let A be any arbitrary symmetric matrix, for example a 2x2 matrix with entries
a b b a

(a b)
(b a), this that matrix multiplied on the right of 0 1 1 0, = (b a)
(a b) , also a symmetric matrix, and therefore this matrix is also an element of W, this W is a T-invariant subspace.

but the back of the book does not say it is T-invariant

please point out if i am making a mistake
test tomorrow

thanks!

 

[stringy]

Remember, a symmetric matrix A satisfies $A=A^T$. That places no restriction on the main diagonal entries. An arbitrary symmetric 2x2 should be of the form

$$\begin{bmatrix} x & y \\ y & z \end{bmatrix}$$