List of top Mathematics Questions on Linear Algebra asked in AP PGECET

Consider the system of equations:
\[ x + 2y - z = 3 \\ 2x + 4y - 2z = 7 \\ 3x + 6y - 3z = 9 \]
Which of the following statements is true about the system?
Determine the value of $\lambda$ and $\mu$ for which the system of equations
$x + 2y + z = 6$,
$x + 4y + 3z = 10$,
$2x + 4y + \lambda z = \mu$
has a unique solution.
If the matrix \[ A = \begin{pmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{pmatrix} \] has three distinct eigenvalues, and one of its eigenvectors is \[ \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, \] then which of the following can be another eigenvector of \( A \)?
If the systems of equations $3x - 2y + z = 0$, $5x + ay + 15z = 0$, $x + 2y - 3z = 0$ have non-zero solution, then $a =$ ...............
If \( A = \begin{pmatrix} 2 & -1 \\ 3 & 2 \end{pmatrix} \) is a \( 2 \times 2 \) matrix, then the eigenvalues of the matrix \( 2A^2 - 4A + 5I \) are ........., where \( I \) is the \( 2 \times 2 \) unit matrix.
If $(a, b, c)$ is the unique solution of the system of linear equations
$x + y + z = 2, 2x + y - z = 3, 3x + 2y + z = 4$,
then $b^2 + c^2 = ........$
If the matrix \( A = \begin{pmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{pmatrix} \) has three distinct eigenvalues and one of its eigenvectors is \( \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} \), then which of the following can be another eigenvector of \( A \)?
Determine the value of $\lambda$ and $\mu$ for which such that the system of equations $x + 2y + z = 6$, $x + 4y + 3z = 10$, and $2x + 4y + \lambda z = \mu$ has infinite number of solutions.
If \( A \) is a 3 \(\times\) 3 matrix with eigenvalues 1, 2, and 3, what is the trace of \( A^2 - 3A + I \)?
If \( A = \begin{pmatrix} -3 & 2 \\ 1 & 0 \end{pmatrix} \) is a \( 2 \times 2 \) matrix, then \( A \) satisfies the relation ...........
Let \( i \) be an imaginary number such that \( i = \sqrt{-1} \). Let \( a \) and \( b \) be real numbers satisfying \( a^2 + b^2 = 1 \).
Then, the eigenvalues of the matrix \[ \begin{bmatrix} -a & b \\ b & a \end{bmatrix} \] are ...........