List of top Mechanical Engineering Questions on Linear Algebra

The system of equation \(x - y + 3z = 0, x + z = 0, x + y - z = 0\) has _______

Let \(A = \begin{pmatrix} 2026 & 2025 & 2024 \\ 2025 & 2024 & 2023 \\ 2024 & 2023 & 2023 \end{pmatrix}\) be a matrix. Then, \(\det(A)\) is equal to _______

If a \(2 \times 2\) matrix \(A\) has eigenvalues 1 and 4 with the corresponding eigenvectors \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\), respectively, then \(A\) is ________

Let \(A = \begin{pmatrix} 2026 & 2025 & 2024 \\ 2025 & 2024 & 2023 \\ 2024 & 2023 & 2023 \end{pmatrix}\) be a matrix. Then, \(\det(A)\) is equal to _______

If a \(2 \times 2\) matrix \(A\) has eigenvalues 1 and 4 with the corresponding eigenvectors \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\), respectively, then \(A\) is ________

The system of equation \(x - y + 3z = 0, x + z = 0, x + y - z = 0\) has _______

Assertion (A): If \(P = \begin{bmatrix}1 & 4 \\ 2 & 3 \end{bmatrix}\), then \(P^2 - 4P - 5I = 0\).
Reason (R): Every square matrix satisfies its own characteristic equation.

Assertion (A): Any square matrix \(P\) and its transpose \(P^T\) have the same eigen values.
Reason (R): The eigen values of an idempotent matrix are either zero or unity.

If \( A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \) satisfies the matrix polynomial equation \( A^2 - 4 + kI_2 = 0 \), then determine the value of \( k \).

The solution of the following system of linear equations
\(4x_1 - 8x_2 - 2x_3 = 0\)
\(3x_1 - 5x_2 - 2x_3 = 0\) 
\(2x_1 - 8x_2 + x_3 = 0\) is:

The eigenvalues of the matrix
\[ \begin{bmatrix} 1 & -2 & 3 \\ -2 & 2 & -4 \\ 3 & -4 & 7 \end{bmatrix} \] are: