1. (a) Find the row-reduced echelon form of A =7 1 2 3 4 5 6 a, showing all elementary row operations used. (b) Is there a simpler way of ﬁnding the row-reduced echelon form of B = 7 1 2 4 5 6 3 2 1 without doing all the painfulcalculations? Please explain. 2. Let A =2 1 3 5. Find all 2 × 2 matrices B such that AB = BA. Suggestion: Set B =x y z w. Then AB = BA gives a system of equations for the coeﬃcients x,y,z,w. Your ﬁnal answer will be of the form B = aX + bY , where X,Y are speciﬁc 2 × 2 matrices and a,b ∈ R are arbitrary.

3. We deﬁne the trace of an n × n matrix B = (bij) by the formula tr(B) =

n P i=1

bii.

(a) Is it possible for a 3×3 invertible matrix to have trace 0? If so, give an example. If not, brieﬂy explain why no such matrix exists. (b) Give an example of a noninvertible 3 × 3 matrix with all distinct non-zero entries and trace 0. 4. (a) Calculate the determinant of A = a + 1 2 1 −1 1 a + 1 −1 2 2 −1 a + 1 1 −1 1 2 a + 1 . For which real numbers a is A invertible? (b) Find all real eigenvalues of A. That is, ﬁnd all real roots λ of the equation 0 = |λI4 − A|. (c) For which a is tr(A) = 1? 5. Let X = x y z . Find all eigenvalues of the 3 × 3 matrix A = XXT.