Stetson University Linear Algebra Matrix Problem

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.

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