Matrices

Construct an m × n matrix A = [a_ij], where a_ij is given by

a_ij = `("i" - 2"j")^2/2` with m = 2, n = 3

Construct an m × n matrix A = [a_ij], where a_ij is given by

a_ij = `|3"i" - 4"j"|/4` with m = 3, n = 4

Find the values of p, q, r, and s if

`[("p"^2 - 1, 0, - 31 - "q"^3),(7, "r" + 1, 9),(- 2, 8, "s" - 1)] = [(1, 0, -4),(7, 3/2, 9),(-2, 8, -pi)]`

Determine the value of x + y if `[(2x + y, 4x),(5x - 7, 4x)] = [(7, 7y - 13),(y, x + 6)]`

Determine the matrices A and B if they satisfy 2A – B + `[(6, - 6, 0),(- 4, 2, 1)]` = 0 and A – 2B = `[(3, 2, 8),(-2, 1, -7)]`

If A = `[(1, "a"),(0, 1)]`, then compute A⁴ 

Consider the matrix A_α = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Show that `"A"_alpha "A"_beta = "A"_((alpha + beta))`

Consider the matrix A_α = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Find all possible real values of α satisfying the condition `"A"_alpha + "A"_alpha^"T"` = I