Usually in this course, our matrices only have real entries. For this exercise specifically, you're going to think about matrices that have complex entries. (You'll do a bit more of this in your Week 6 tutorials, in the context of eigenvalues and characteristic polynomials.) Let A be the following matrix. A = z + 4i 3z -2 0 1 -i 1 i 0 (a) Find all complex numbers z such that det(A) = i or det(A) = -i (there is one value of z that works for each). Show all of your work. (b) Fill in the value of z that results in the determinant being i from above, to obtain a new matrix B with no variables in it. Find B-1. Show all of your work.
Usually in this course, our matrices only have real entries. For this exercise specifically, you're going to think about matrices that have complex entries. (You'll do a bit more of this in your Week 6 tutorials, in the context of eigenvalues and characteristic polynomials.) Let A be the following matrix. A = z + 4i 3z -2 0 1 -i 1 i 0 (a) Find all complex numbers z such that det(A) = i or det(A) = -i (there is one value of z that works for each). Show all of your work. (b) Fill in the value of z that results in the determinant being i from above, to obtain a new matrix B with no variables in it. Find B-1. Show all of your work.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Usually in this course, our matrices only have real entries. For this exercise specifically, you're going to think
about matrices that have complex entries. (You'll do a bit more of this in your Week 6 tutorials, in the
context of eigenvalues and characteristic polynomials.)
Let A be the following matrix.
A =
z + 4i
3z
-2
0 1
-i 1
i 0
(a) Find all complex numbers z such that det(A) = i or det(A) = -i (there is one value of z that works for
each). Show all of your work.
(b) Fill in the value of z that results in the determinant being i from above, to obtain a new matrix B with
no variables in it. Find B-¹. Show all of your work.
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