1300-A2aq - jsicbiyhc PDF

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MATH 1300 Problem Set 2 February 16, 2021

Due: February 22, 2021, 10am Be prepared to upload your solution to each question on a separate page. We mark for process as well as for accuracy; always show your work, and give explanations where appropriate. Part marks may be available even if computation errors are made. In particular, in any row-reduction we expect you to indicate the elementary row-operations used; without that we will not attempt to search for part marks.

[3+4]

Question 1. (a) For a square invertible matrix A , let A∗ = (A−1 )T . Show that for square invertible matrices of the same size (shape), (AB)∗ = A∗ B ∗ . (b) Let A , B , C , and D be invertible square matrices of the same size (shape). Solve for D in terms of A , B , and C and their inverses or tranposes if C T ((A−1 D)T )−1 B = AT Your answer should not contain any nested expressions in parentheses.

[2]

[3]

Question 2.

Question 3.

Write the 4 × 4 matrix of entries [aij ] with aij =

2

Let p(x) = x − x − 1 and A =



0 1 1 1



.

Evaluate p(A) .

[2+4]

Question 4.

Let A =



5 3 2 1



.

(a) Find A−1 by any method. (b) Using your answer to part(a), find a matrix X so that   −1 0 XA = B =  1 −8  0 3 1

i i + 2j

Let B =



 2 4 . 2 7

[7]

Question 5.

[3]

(a) Reduce B to the identity matrix by elementary row operations

[2]

(b) Using your solution to (a), write an expression for B −1 as a product of elementary matrices. [You do not need to actually calculate B −1 .)

[2]

(c) Using your solution to (b), write an expression for B as a product of elementary matrices.

[5]

Question 6.

 1 1 a Let M =  1 b b  . c c c 

Find all possible combinations of values of a , b , and c so that M is invertible. [Use row reduction, and your understanding of how a row echelon matrix will establish that M is not invertible.]

[30]

TOTAL

2...