v = [ 2 , 2 , 3 ] and w = [ -2 , -1 , 2 ]. What is 3(2v -w)?

Let A = [ 4 , 2 , -1 ; 0 , 0 , 1] and B is a matrix such that it is possible to form product AB. How many rows does B have?

Express 4/ (√3 + i ) in cartestian form.

Let v = [ 1 , 3 , -2 , -1 ] and w = [ -2 , 1 , k , -3 ]. For which value of k are the vectors v and w orthogonal?

If A is an n x n matrix and B is an n x p matrix such that AB = 0, then B = 0.

Give a unit vector with the same direction as v = [ 3 , 1 ]

Let A = [ 5 , -3 ; 2 , 3 ] and B = [ -1 , -1 ; 2 , 1 ]. What is AB?

Is v an eigenvector of A = [ 2 , 0 , 0 ; -1 , 2 , 1 ; 1 , 1 , 2]? If it is, determine the correct corresponding eigenvalue.

For what value of k does the following linear system have no solution? [kx + y + z = 1] [ x + ky + z = 1] [ x + y + kz = 1 ]

If A is an n x n matrix, and rank(A) = n, then Det(A) = 0

Find all values of "a" such that [ 1 , 2 , a ] is a linear combination of [ 0 , 2 , 3 ] and [ -1 , -2 , 5 ]

Let C = [ 0 , 4, 0 ; 1 , 3 , 2 ; 4 , -3 , -1 ] What is det(C)?

Let A = [ 4 , 0 , 0 ; 7 , 2 , -2 ; 1 , -2 , 2 ]. Find the eigenvalues of A and their algebraic multiplicities.

Suppose that A is a 4 x 8 matrix with rank 2. What is the dimension of null(A)?

If A is a square matrix, then AA^T and A^T are orthogonally diagonalizable.

What is the angle between v = [ 1 , 2 , 2 ] and w = [ -3 , 0 , 1 ]

Suppose A is an n x n matrix. Write 5 statements which are equivalent to " A is invertible"

Let A = [ -3 , -4 , -5 ; -1 , 0 , 5 ; 1 , 6 , 1 ]. You may use, without proof, the fact that the eigenvalues of A are -4 with algebraic multiplicity 2 and 6, with algebraic multiplicity 1. Show that the eigenspace of -4 is equal to 1.

Suppose A and B are 5 x 5 matrices with det(A) = 2 and det(B) = -4. Calculate det(inverse of A (B^3))

If A is a 4 x 3 matrix and nullity (A^T) = 2, then rank(A) = 2

Do the solutions to the following system of equations form a point, line, plane or all of r3? [ 2x - y = 2] [ x + 2y - z = 1] [-5y + 2z = 0 ]

Let B = [ 1 , -1 , 0 ; 0 , 1 , 2 ; 0 , 0 , 1]. Determine whether or not B is invertible. If it is, calculate B inverse.

Find all values of "a" for which the matrix A = [ 4 , 2a-10 ; 0 , 4]

Let A = [ 2 , 1 , 2 ; 0 , 3 , 2 ; 0 , 0 , 5]. Give an eigenvector for lambda = 5.

If v is both in the row space and in the column space of a square matrix A, then v = 0