# College Paper Lab

liner algebra- College Paper Lab | collegepaperslab.com

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 ï¬nding the row-reduced echelon form of B =ï£« ï£­ 7 1 2 4 5 6 3 2 1 ï£¶ ï£¸without doing all the painfulcalculations? Please explain. 2. Let A = 2 1 3 5 . Find all 2 Ã— 2 matrices B such that AB = BA. Suggestion: Set B = x y z w . Then AB = BA gives a system of equations for the coeï¬ƒcients x,y,z,w. Your ï¬nal answer will be of the form B = aX + bY , where X,Y are speciï¬c 2 Ã— 2 matrices and a,b âˆˆ R are arbitrary.

3. We deï¬ne the trace of an n Ã— n matrix B = (bij) by the formula tr(B) =

n P i=1

bii.

(a) Is it possible for a 3Ã—3 invertible matrix to have trace 0? If so, give an example. If not, brieï¬‚y explain why no such matrix exists. (b) Give an example of a noninvertible 3 Ã— 3 matrix with all distinct non-zero entries and trace 0. 4. (a) Calculate the determinant of A = ï£« ï£¬ ï£­ a + 1 2 1 âˆ’1 1 a + 1 âˆ’1 2 2 âˆ’1 a + 1 1 âˆ’1 1 2 a + 1 ï£¶ ï£· ï£¸ . For which real numbers a is A invertible? (b) Find all real eigenvalues of A. That is, ï¬nd all real roots Î» of the equation 0 = |Î»I4 âˆ’ A|. (c) For which a is tr(A) = 1? 5. Let X =ï£« ï£­ x y z ï£¶ ï£¸. Find all eigenvalues of the 3 Ã— 3 matrix A = XXT.