List of top Engineering Mathematics Questions on Linear Algebra asked in TS PGECET

Determine the rank of the matrix:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \]
Find the inverse of the matrix:
\[ \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \]
Find the eigenvalues of the matrix: \[ A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} \]
Which of the following is a valid method for solving a system of linear equations?
$L$ is a lower triangular matrix with all Principal diagonal elements equal to 1 and $U$ is an upper triangular matrix such that \[ LU = \begin{bmatrix} 1 & 3 & 0 \\ 3 & 7 & 1 \\ 2 & 8 & 3 \end{bmatrix}, \] then the Trace of $L$ + Trace of $U$ =
For the matrix \[ \begin{bmatrix} 2 & 1 & 1 \\ 0 & 2 & 1 \\ 1 & 0 & 1 \end{bmatrix}, \] an Eigen vector among the following vectors is
If \( \overline{X}_1 \) and \( \overline{X}_2 \) are any two eigenvectors corresponding to eigenvalues \( \lambda_1 \) and \( \lambda_2 \) respectively, then
If $l>0, K<0$ are roots of $x^2 + x - 20 = 0$, the system of linear equations $3x - y + 4z = 3$, $x + 2y - 3z = -2$ and $6x + 5y + Pz = -3$ has?
Evaluate \( \log_{e^{2000}} e^{2021} \cdot \log_{e^{2021}} e^{2022} \cdot \log_{e^{2022}} e^{2023} \cdot \log_{e^{2023}} e^{2024} \)
If the characteristic polynomial of the matrix \( A_{3\times3} \) is given by \( f(\lambda) = \lambda^3 - 10\lambda^2 + 27\lambda - 18 \), then trace of A and determinant of A respectively are
If the equations $x^2+2x+K=0$ and $x^2+Kx+2=0$ have one root in common, then the product of the other two roots is?
If $$ A = \begin{pmatrix} 3 & -2 \\ 4 & -2 \end{pmatrix} $$ satisfies the matrix equation $$ A^2 - K A + 2I = 0, $$ then the value of $ K $ is:
If $$ A = \frac{1}{2} \begin{pmatrix} -1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 1 \\ 1 & 1 & -1 & 1 \\ 1 & 1 & 1 & -1 \end{pmatrix}, $$ then the inverse of $ AA^T $ is:
If the characteristic equation of a $3 \times 3$ square matrix A is $ax^3+bx^2+cx+d=0$, $(a \neq 0)$ then the Trace of A + det A =
If $\lambda_1$ and $\lambda_2$ are two distinct eigen values of a symmetric matrix, $X_1$ and $X_2$ are the eigen vectors corresponding to $\lambda_1, \lambda_2$ respectively, then
If the Eigenvalues of Skew-Hermitian matrices and Eigenvalues of Hermitian matrices are plotted on Argand plane, then the number of points having amplitude $\frac{7\pi}{4}$ is
Let $A$ be a $3 \times 3$ matrix. If $\lambda_1, \lambda_2, \lambda_3$ are the eigenvalues of $A$, then the eigenvalues of the matrix $(I + aA)^{-1}(I + bA)$, where $a, b$ are scalars such that $a \lambda_i \neq -1$ and $b \lambda_i \neq -1$ for $i = 1, 2, 3$, are:
Consider the linear system of non homogeneous equation in three variables AX = B. $\rho(A)$ and $\rho([A:B])$ are roots of the equation $x^2+bx+c=0$. AX = B has infinite number of solutions, then the option containing two possible pairs of values of (b,c) is (Note: $\rho$ denotes rank.)
A and B are two nonsingular matrices. If the characteristic equation of A is $ a_0\lambda^3 + a_1\lambda^2 + a_2\lambda + a_3 = 0 $ and characteristic equation of $ B^{-1}AB $ is $ b_0\lambda^3 + b_1\lambda^2 + b_2\lambda + b_3 = 0 $, then