If A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], find AB and BA and state whether they are equal.

Define a symmetric and a skew-symmetric matrix, and state what can be said about the diagonal of a skew-symmetric matrix.

If A is a 3 × 3 matrix with A² = A and A is not the identity, what is such a matrix called?

Prove that (AB)^T = B^TA^T for matrices for which the product is defined.

If A = [[2, 3], [1, 4]], find A⁻¹.

Find the projection of a = 2i + 3j + 2k on b = i + 2j + k.

Evaluate the definite integral of sin x from 0 to π.