1. Given that the determinant of the matrix A is -12, what is the determinant of matrix B? \[ A = \begin{pmatrix} 1 & 3 & 0 \\ 2 & 6 & 4 \\ -1 & 0 & 2 \end{pmatrix} \], \[ B = \begin{pmatrix} 2 & 6 & 0 \\ 4 & 12 & 8 \\ -2 & 0 & 4 \end{pmatrix} \]
2. A real (4x4) matrix A satisfies the equation A² = I, , where I is the (4 x4) identity matrix. The positive eigenvalue of A is______.
3. For matrices M, N and scalar C, which one of the following properties does NOT always hold?
4. Consider the below matrix A. Which is obtained by reversing the order of the columns of the identity matrix I₆. Let P = I₆ + α * J₆, where α is non-negative real number. The value of α for which det(P) = 0 is _____. \[ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix} \]
5. Given the system of equations:
x + 2y + 2z = b₁
5x + y + 3z = b₂
Which of the following is true regarding its solutions?
6. A system matrix is given as follows. The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is ___. \[ A = \begin{pmatrix} 0 & 1 & -1 \\ -6 & -11 & 6 \\ -6 & -11 & 5 \end{pmatrix} \]
7. If 1, 0, and -1 are the eigenvalues of a 3×3 matrix A, then the trace of A^2 + 5A is equal to ___
8. For the matrix,
\[
M = \begin{pmatrix}
1 & 0 & -1 \\
0 & 1 & -1 \\
1 & 1 & -2
\end{pmatrix}
\]
consider the following statements:
(P) the characteristic equation of M is λ^3-λ =0.
(Q) M^-1 Does not exist.
(R) The matrix M is Diagonalizable.
9. Let A be an (m×n) matrix and B be an (n×m) matrix. It is given that determinant(Im + AB) = determinant(In + BA), where Ik is the k×k identity matrix. Using the above property, the determinant of the matrix below is ___
10. Let r and s be real numbers. If then the system of linear equations AX = b has \[ A = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 3 \\ r & s & 0 \end{pmatrix} \] and, \[ b = \begin{pmatrix} 1 \\ 1 \\ s-1 \end{pmatrix} \]
11.
Matrix A and one of its normalized eigenvectors:
A = \[ \begin{pmatrix} 5 & 3 \\ 1 & 3 \end{pmatrix} \]
One of the normalized Eigen vectors is given as:
12.
x + 2y + z = 4
2x + y + 2z = 5
x - y + z = 1
The system of algebraic equations given above has ___ solutions:
13. Given that \[ A = \begin{pmatrix} -5 & -3 \\ 2 & 0 \end{pmatrix} \] and \[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \] the value of A^3 is
14. The minimum eigenvalue of the following matrix is \[ \begin{pmatrix} 3 & 5 & 2 \\ 5 & 12 & 7 \\ 2 & 7 & 5 \end{pmatrix} \]
15. A matrix has eigenvalues -1 and -2. The corresponding eigen vectors are \[ \begin{pmatrix} 1 \\ -1 \end{pmatrix} \] and \[ \begin{pmatrix} 1 \\ -2 \end{pmatrix} \] respectively. The matrix is?