List of practice Questions

Matrix A has the eigenvalues 1, 2, and 3. The Trace of $A^2$ is

For the matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, if the relation $a+b=c+d$ holds and $a,b,c,d \neq 0$, then which one of the following statements about $A$ is FALSE?

Let $AX=B$ be a system of n-linear equations in n unknowns then

Let $A$ be a $2\times2$ real symmetric matrix such that the trace of $A$ is $6$ and the determinant of $A$ is $5$. Which one of the following statements is true?

If a $3 \times 3$ matrix $A$ has eigenvalues $2, 3, -4$, find $\det(A)$.

Find the inverse of the given matrix: \[ A = \begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix} \]

Let $ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix} $. If the eigenvalues of $ A $ are -2, 1, and 2, then the value of $ k $ is __________. (Answer in integer)

Let $ A $ be a $ 2 \times 2 $ matrix as given:

\[ A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \]
What are the eigenvalues of the matrix $ A^{13} $?

The eigenvalues of the matrix \[ A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix} \] are $\lambda_1, \lambda_2, \lambda_3$. The value of $\lambda_1 \lambda_2 \lambda_3 (\lambda_1 + \lambda_2 + \lambda_3)$ is:

The eigenvalues of the matrix $ A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $ are:

The eigenvalues of the matrix

are $ \lambda_1, \lambda_2, \lambda_3 $. The value of $ \lambda_1 \lambda_2 \lambda_3 ( \lambda_1 + \lambda_2 + \lambda_3 ) $ is:

The eigenvalues of the matrix: \[ \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \] are ________

Let \[ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix}. \] If the eigenvalues of $ A $ are -2, 1, and 2, then the value of $ k $ is _.
(Answer in integer)

The eigenvalues of the matrix
\[ \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \]
are _______ .

The eigenvalues of the matrix \[ \begin{bmatrix} 1 & 2
0 & 3 \end{bmatrix} \] are \_\_\_\_\_ .

The sum of the eigenvalues of the matrix
$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2 $
is _____________ (rounded off to the nearest integer).

What are the eigenvalues of the matrix $\begin{bmatrix} 2 & 1 & 1 \\[0.3em] 1 & 4 & 1 \\[0.3em] 1 & 2 & 2 \end{bmatrix}$?

The matrix $\begin{bmatrix} 1&a\\8&3 \end{bmatrix}$(where a > 0) has a negative eigenvalue if a is greater than

Consider the matrix \[A = \begin{bmatrix} 5 & -4 \\ k & -1 \end{bmatrix},\]
where $k$ is a constant. If $\det(A) = 3$, then the ratio of the largest eigenvalue of $A$ to $k$ is ___________ (rounded off to 1 decimal place).

The values of $ \mu $ which satisfy the equation $ A^{100} \vec{x} = \mu \vec{x} $, where $ A = \begin{bmatrix} 2 & -1 \\ 0 & -2 \\ 1 & 1 \end{bmatrix} $ are

The matrix $ A = \begin{bmatrix} 3 & -1 & 1 \\ 1 & -5 & 1 \\ 1 & -1 & 3 \end{bmatrix} $ has eigenvalues 2, 3, 6 then the eigenvalues of $ A^4 $ are

The sum of the eigenvalues of the matrix $ A = \begin{bmatrix} 1 & 2
3 & 4 \end{bmatrix}^2 $ is \_\_\_\_\_ (rounded off to the nearest integer).

If $M$ is an arbitrary real $n \times n$ matrix, then which of the following matrices will have non-negative eigenvalues?

For the matrix \[ [A] = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 3 & 1 & 2 \end{bmatrix} \] which of the following statements is/are TRUE?