I need someone to solve these 5 questions with support answers step by step. Look down to the questions. Please need it in word file

1. (a) Find the row-reduced echelon form of A=

7 1 2 3

4 5 6 a

3), 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 painful

calculations? Please explain.

2

2. Let A=

Find all 2 x 2 matrices B such that AB = BA. Suggestion: Set B =

3

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 x 2 matrices and a, b e R are arbitrary.

i=1

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

(a) Is it possible for a 3 x 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 x 3 matrix with all distinct non-zero entries and trace 0.

a +1 2 1 – 1

1

4. (a) Calculate the determinant of A=

a +1

-1 2

. For which real numbers a is A invertible?

2 -1 a +1 1

-1 1 2

a +1

(b) Find all real eigenvalues of A. That is, find all real roots of the equation 0 = [X14 – A.

(c) For which a is tr(A) = 1?

2

5. Let X=

Y

Find all eigenvalues of the 3 x 3 matrix A = XXT.