List of top Engineering Mathematics Questions on Matrices asked in TS PGECET

The system of equations \[ x+4y+6z=20,\qquad x+y+z=6,\qquad x+\lambda y+6z=20 \] has
Let \(A\) be a real symmetric matrix. \(\lambda_1,\lambda_2\;(\lambda_1\neq\lambda_2)\) be two eigen values of \(A\) and \(X_1,X_2\) are respectively the eigen vectors of \(A\) corresponding to \(\lambda_1\) and \(\lambda_2\), then \(X_1^TX_2=\)
\[ \lim_{n\rightarrow\infty}\frac{5n^{2}+4}{7n^{2}+6n}= \]
If \[ f(x,y,z,w)=x^{2}e^{2y+3z}\cos(4w), \] then \[ \frac{\partial f}{\partial z} \] at \((2,0,-2,1)\) is
If \[ \begin{bmatrix} 1 1 \end{bmatrix} \] is an eigenvector of the matrix \[ A= \begin{bmatrix} m& 3 \\ 3& m \end{bmatrix}, \] then the corresponding eigenvalue is:
The system of equations \[ x-2y+3z=6,\quad 3x+y-4z=-7,\quad 5x-3y+2z=5 \] has:
Which of the following cannot be an eigen value for an unitary matrix?
If \(A\) is symmetric, \(P\) and \(B\) are skew symmetric matrices, then which of the following is correct?
Let \(A=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\). If \(u_1\) and \(u_2\) are column matrices such that \(Au_1 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\) and \(Au_2 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}\), then \(u_1 - u_2\) is
If \( A = \begin{pmatrix} 2024 & 2021 \\ 2023 & 2022 \end{pmatrix} \), then the value of \[ \left| A^{2024} - A^{2023} \right| \] is
If \( P = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \) is the modal matrix of \( A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{pmatrix} \) then the sum of all the elements of \( P^{-1}AP \) is
If \( \lambda_1, \lambda_2, \lambda_3 \) are the eigenvalues of the matrix \( A = \begin{pmatrix} 1 & 0 & 2 \\ 0 & -3 & 1 \\ 0 & 0 & 4 \end{pmatrix} \), then the values of \[ (\lambda_1 + \lambda_2 + \lambda_3) \ \text{and} \ (\lambda_1 \cdot \lambda_2 \cdot \lambda_3) \] are respectively
Let \[ A = \begin{bmatrix} 1 & -5 & 2 \\ 7 & 0 & 6 \\ 5 & -3 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 6 & 7 \\ 3 & 8 & 4 \\ 7 & 6 & 1 \end{bmatrix}. \] If \( C \) and \( D \) are two \( 3 \times 3 \) matrices such that: - \( A + C \) is symmetric, - \( A - C \) is skew-symmetric, and - \( B = D^\top \), then find \( (C - D)^\top \).
The eigenvalues of a $3 \times 3$ matrix A are 1, 3, 7 then the eigenvalues of $\text{adj}(A)$ are
If the linear transformation \( X = BY \) transforms \( X^\top AX \) to \( Y^\top PY \), then \( P = \):
Let \( A = \begin{bmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{bmatrix} \), \( P = \begin{bmatrix} -1 & 1 & 1 \\ 0 & -1 & 2 \\ 1 & 1 & 1 \end{bmatrix} \). Then \( \det(P^{-1} A P - 2I) \) is equal to:
If \( X_1 = \begin{bmatrix} 1 \\ i \end{bmatrix} \), \( X_2 = \begin{bmatrix} i \\ 1 \end{bmatrix} \) are given vectors and \( A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \).
If \( P = [X_1, X_2] \), then \( P^{-1} A P \) is:
The rank of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & -4 & -8 & 3 \\ 2 & 4 & 3 & 2 \\ 0 & -4 & -11 & 5 \end{bmatrix} \) is:
The rank of the matrix A = $ \begin{bmatrix} 1 & 3 & 4 & 3 \\ 3 & 9 & 12 & 9 \\ -1 & -3 & -4 & -3 \end{bmatrix} $ is