1. ## Diagonalizable Matrix

1.) Is the matrix A = [[3,1],[0,3]] diagonalizable? Explain.

Well, I know a matrix is said to be diagonlizable if A is similar to a digonal matrix, that is, if A = PDP^(-1) for some invertible matrix P and some diagonal matrix D; also, it is diagonalizable iff A has n linearly independent eigenvectors;

So I know (A - LI)x = 0;

How do I know the eigenvalues though and how would I show its diagonalizable?

2.) Suppose we have an n x n matrix A with det(A^4) = 0. Can the matrix A be invertible; if so, give an example of such a matrix A. If not, show whyy Matrix A cannot be invertible.

3.) Let lambda (L) be an eigen value of an invertible Matrix, A. Show that L^2 is an eigen value of A^(2).

2. Originally Posted by fifthrapiers
1.) Is the matrix A = [[3,1],[0,3]] diagonalizable? Explain.

Well, I know a matrix is said to be diagonlizable if A is similar to a digonal matrix, that is, if A = PDP^(-1) for some invertible matrix P and some diagonal matrix D; also, it is diagonalizable iff A has n linearly independent eigenvectors;

So I know (A - LI)x = 0;

How do I know the eigenvalues though and how would I show its diagonalizable?
The solution of that equation for a general x is for $\displaystyle det(A - LI) = 0$. So:
$\displaystyle \left | \begin{array}{cc} 3 - L & 1 \\ 0 & 3 - L \end{array} \right | = (3 - L)^2 - 1 \cdot 0 = 0$

Now solve for L to give your eigenvalues. If this equation has no solutions then you know the matrix is not diagonalizable.

-Dan

3. Originally Posted by fifthrapiers
2.) Suppose we have an n x n matrix A with det(A^4) = 0. Can the matrix A be invertible; if so, give an example of such a matrix A. If not, show whyy Matrix A cannot be invertible.
$\displaystyle det(A^4) = det(A \cdot A \cdot A \cdot A) = det(A) \cdot det(A) \cdot det(A) \cdot det(A) = \left ( det(A) \right )^4 = 0$

Thus det(A) = 0. Thus A is not invertible.

-Dan