Algebra Matrices Mock Tests

Let A be a 3x3 matrix with eigenvalues 1, 2, and 3. Which of the following is/are TRUE?

Let ω be a complex cube root of unity (ω ≠ 1). The determinant |1 ω ω²; ω ω² 1; ω² 1 ω| equals:

Let A be a 2x2 matrix satisfying A² - 4A + I = O. If tr(A) = 4, then det(A) equals:

Let S = {1, 2, 3, ..., 10}. The number of 2×2 matrices with distinct entries from S such that the sum of each row equals 5 and the sum of each column equals 5, is

The number of 3 x 3 symmetric matrices with entries from {0, 1} is:

Let T = {0, 1}. The number of 3 x 3 matrices with entries from T such that the sum of each row equals the sum of the corresponding column (R_i = C_i for i = 1, 2, 3) is equal to:

Let S = {x ∈ ℤ : −1 ≤ x ≤ 1}. The number of 3 × 3 matrices with entries from S such that the sum of each row equals the sum of the corresponding column (i.e., R_i = C_i for i = 1, 2, 3) is:

Let A be a 3x3 matrix with real entries such that A + A^T = 0 (skew-symmetric) and det(A) = 0. If the entries of A are chosen from {-1, 0, 1}, the number of such matrices A is