If the matrix A = ⌈ 1 0 0 ⌉ ⌈ 0 2 0 ⌉ ⌊ 3 0 -1 ⌋ satisfies the equation A20 + αA19 + βA = ⌈ 1 0 0 ⌉ ⌈ 0 4 0 ⌉ ⌊ 0 0 1 ⌋ for some real numbers α and β, then β - α is equal to:

Let T2 be the set of all 2 × 2 matrices A = a bc d where a, b, c, d ∈ {0, 1, 2, 3, 4, 5, 6}. Find the number of matrices A such that A is either symmetric or skew-symmetric and det(A) is divisible by 7.

If f(x) = [cos x -sin x 0; sin x cos x 0; 0 0 1], then f(x + y) is equal to:

If A = 13 [1 -2 2; -2 1 2; -2 -2 -1], then matrix A is: