AERO GATE TOPPER
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Linear Algebra
AERO GATE TOPPER
1. Find the rank of the matrix: \[ A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ 1 & 1 & 1 & 1 \\ 3 & 5 & 7 & 9 \end{pmatrix} \]
2. Which of the following is an eigenvalue of the matrix \[ B = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix} \]?
3. How many free variables are present in the system of equations represented by the matrix: \[ \begin{pmatrix} 1 & 2 & 3 & | & 4 \\ 0 & 1 & 4 & | & 5 \\ 0 & 0 & 1 & | & 1 \end{pmatrix} \]
4. A matrix A is diagonalizable if:
5. Calculate the determinant of the matrix: \[ C = \begin{pmatrix} 4 & 1 & 2 \\ 0 & 3 & 5 \\ 1 & 0 & 6 \end{pmatrix} \]
6. Find the geometric multiplicity of the eigenvalue λ = 3 for the matrix: \[ A = \begin{pmatrix} 3 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 3 \end{pmatrix} \]
7. Which of the following matrices is not invertible?
8. Calculate the trace of the matrix: \[ A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{pmatrix} \]
9. Which theorem states that every square matrix satisfies its own characteristic equation?
10. A matrix has 6 columns and a rank of 2. Calculate the nullity of the matrix.
11. Find the determinant of the following matrix: \[ A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \]
12. Which process is used to orthogonalize a set of vectors in an inner product space?
13. Find the rank of the matrix: \[ B = \begin{pmatrix} 1 & 0 & 1 \\ 2 & 0 & 2 \\ 3 & 0 & 3 \end{pmatrix} \]
14. If a matrix has 5 columns and rank 3, what is its nullity?
15. Is the matrix \[ C = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \] invertible? Answer with 1 for yes and 0 for no.
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