List of top Engineering Mathematics Questions on Linear Algebra

Consider the matrix \( A = \begin{pmatrix} 9 & 15 \\ 15 & 50 \end{pmatrix} \). The matrix A is decomposed using Cholesky decomposition. Determine the value of \(l_{22}\).
Consider the homogeneous system of linear equations: \( \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \). What does the solution set of this system represent geometrically?
Let \( P = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} \). Which of the following statements is/are correct?
1. P\(^T\)P = I
2. P is skew-symmetric
3. The value of each eigenvalue of P is 1
4. The trace of P is equal to the sum of its eigenvalues
\( (y + 3x - 13)^3 + (x + y - 7)^2 = 0 \) where x and y are integers. The value of \(x^3 + y^3\) is _____.
Eigenvalue question: \[\begin{pmatrix} 6 & 8 \\ 4 & 2 \end{pmatrix} \quad \text{Eigenvalue options:} \quad \text{(i) 1, (ii) 2, (iii) 3, (iv) 4}\]
Given the matrix equation: \[A = \begin{pmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{pmatrix} \quad A \cdot X = B\]
Where \( A \) is the matrix, \( X \) is the unknown vector, and \( B \) is a constant vector. Solve for \( X \).
Consider the matrix: \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \] The eigenvalues of the matrix are 0.27 and __________ (rounded off to 2 decimal places).
Consider the system of equations: \[ \begin{aligned} x + y + z &= 6 \\ 2x + 2y + 3z &= 13 \\ 3x + 4y + 5z &= 20 \end{aligned} \] Which of the following statements is true about the solution to this system?
If \( A \) and \( B \) are two non-zero square matrices of same size such that the product matrix \( AB \) is a zero matrix, then which of the following must be true?
Find the eigenvalues of the matrix:
Matrix \[ A = \begin{bmatrix} 4 & 1 & 2 \\ 1 & 3 & 0 \\ 2 & 0 & 5 \end{bmatrix} \]

Consider the matrix:

\[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \]

The eigenvalues of the matrix are 0.27 and ____ (rounded off to 2 decimal places).

Given that \( \lambda \) is an eigenvalue of matrix \( A \) with the corresponding eigenvector \( x \), and \( x \) is also an eigenvector of \( B = A - 2I \), find the relationship between \( \lambda \) and the eigenvalue of \( B \).

If $$ A = \begin{bmatrix} 2 & 3 & 4 \\0 & 4 & 2 \\0 & 0 & 3 \end{bmatrix} $$ then the eigenvalues of $ \text{adj}(A) $ are

If A is $$ A = \begin{bmatrix} 8 & 7 \\5 & 6 \end{bmatrix} $$ then the value of $ \text{det}(A^{121} - A^{120}) = ? $

Let A be square matrix of order m, then nullity of A is
The following system has non-trivial solution $ px + qy + rz = 0, qx + ry + pz = 0, rx + py + qz = 0 $ then which one of the following is TRUE?
If the system of equations: \[ 3x + 2y + z = 0, \quad x + 4y + z = 0, \quad 2x + y + 4z = 0 \] is given, then:
Let $\mathbf{x}$ be an $n \times 1$ real column vector with length $l=\sqrt{\mathbf{x}^T\mathbf{x}}$. The trace of the matrix $P=\mathbf{x}\mathbf{x}^T$ is _____________
The state equation of a second order system is $\dot{x}(t)=Ax(t)$, with $x(0)$ as the initial condition. Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of $A$ and $v_1$ and $v_2$ are the corresponding eigenvectors. For constants $\alpha_1$ and $\alpha_2$, the solution $x(t)$ of the state equation is _____________
Let \(\mathbf{v}_1=\begin{bmatrix}1 \\ 2 \\ 0\end{bmatrix}\) and \(\mathbf{v}_2=\begin{bmatrix}2 \\ 1 \\ 3\end{bmatrix}\) be two vectors. The value of the coefficient \(\alpha\) in the expression \(\mathbf{v}_1=\alpha \mathbf{v}_2+\mathbf{e}\), which minimizes the length of the error vector \(\mathbf{e}\), is
Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\{\lambda_k \mid 1 \le k \le n\}$ and $\{\mathbf{v}_k \mid 1 \le k \le n\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where $B=P^{-1}AP$, respectively, are

For the matrix \[ A = \begin{bmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{bmatrix} \] which of the following statements is/are TRUE? 

Two vectors \([2 \, 1 \, 0 \, 3]^T\) and \([1 \, 0 \, 1 \, 2]^T\) belong to the null space of a \(4 \times 4\) matrix of rank 2. Which one of the following vectors also belongs to the null space?
Two vectors \([2 \, 1 \, 0 \, 3]^T\) and \([1 \, 0 \, 1 \, 2]^T\) belong to the null space of a \(4 \times 4\) matrix of rank 2. Which one of the following vectors also belongs to the null space?
Two vectors \([2 \, 1 \, 0 \, 3]^T\) and \([1 \, 0 \, 1 \, 2]^T\) belong to the null space of a \(4 \times 4\) matrix of rank 2. Which one of the following vectors also belongs to the null space?