University of The District of Columbia Differential Equations Homework

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.