Let $ A$ and $ B$ be $n \times n$ matrix.
which one of the following statement are/is True $?$
$a)$ if $A^n = 0$ for some $ n $, then $\det A = 0$.
$b)$ if A and B have the same characteristic polynomial, then they are similar.
$c)$ if the eigenvalues of $A$ are $\lambda_1,\lambda_2,\lambda_3,\dots,\lambda_n$, then A is similar
to the
diagonal matrix diag($\lambda_1 ,\lambda_2,\lambda_3,\dots,\lambda_n$)
My answer : all options a, b and c..
For option a), take $A = \begin{bmatrix} 0 & 1 \\ 0& 0 \end{bmatrix}$
For option b), take $A = \begin{bmatrix} 0 & 1 \\ 1& 0 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 0& 1 \end{bmatrix}$
For option c), take $A = \begin{bmatrix} 1 & 0 \\ 0& 1 \end{bmatrix}$
Is it True ??….
Best Answer
(a) is true for a nilpotent matrix has all eigenvalues zero. Since det$A$ is equal to the product of eigenvalue, det$A=0$.
(b) is not true. Take $A = \begin{bmatrix} 0 & 1 \\ 0& 0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 0 \\ 0& 0 \end{bmatrix}$
Both have the same characteristics polynomial but $A$ and $B$ are not similar.
(c) is not true in general. It is true when $A$ is a diagonalizable matrix having eigenvalues $\lambda_1, \lambda_2,...,\lambda_n$.