If $A$ is a $3\times 3$ non-zero matrix such that $A^2=0$, then the number of non-zero eigen values of $A$ is:

Question

Let $ A = \begin{bmatrix} 1 & \sqrt{2} \\ -2 & 0 & 1 \end{bmatrix} $ and $ P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $, $ \theta > 0 $. If $ B = P A P^T $, $ C = P^T B P $, and the sum of the diagonal elements of $ C $ is $ \frac{m}{n} $, where $ \gcd(m, n) = 1 $, then $ m + n $ is:

Answer 1

Step 1: $A^2=0$ with $A \neq 0$ means the minimal polynomial $m(x)$ of $A$ divides $x^2$ but is not constant and is not $x$ alone, so $m(x) = x^2$. This makes $A$ a nilpotent matrix.
Step 2: For any nilpotent matrix, every eigenvalue must be 0, so the characteristic polynomial of the $3\times 3$ matrix $A$ is $x^3$.
Step 3: The roots of $x^3=0$ are $\lambda=0,0,0$, i.e. all three eigenvalues (with multiplicity) are zero. Consequently $\text{trace}(A)=0$ and $\det(A)=0$ too, consistent with $A$ being singular and nilpotent.
Step 4: So none of the eigenvalues of $A$ is non-zero.
\[\boxed{0}\]

If \(A\) is a \(3\times 3\) non-zero matrix such that \(A^2=0\), then the number of non-zero eigen values of \(A\) is:

Show Hint

The Correct Option is A

Solution and Explanation

Top Questions on Matrix Algebra

Questions Asked in CPET exam