(a) Suppose that B and C are invertible 4 × 4 matrices satisfying the following: • Columns 1 and 2 of B-¹ are, respectively, 2 -2 1. -2 - and = 3 -2 .Rows 3 and 4 of C-1 are, respectively, [o - - -2 0 1] and [1 1 −1 −1]. If X BC, find the (3, 1)-entry of X-¹ and the (2, 4)-entry of (XT)−¹. Explain your answers fully. (b) Let A and B be n × n matrices, and let y be a non-zero vector in R”. Show that if A - B is invertible, then the equations Ax=y and Bx = y do not have a common solution x ER". Hint: Suppose that x is some solution to both equations, and aim for a contradiction.
(a) Suppose that B and C are invertible 4 × 4 matrices satisfying the following: • Columns 1 and 2 of B-¹ are, respectively, 2 -2 1. -2 - and = 3 -2 .Rows 3 and 4 of C-1 are, respectively, [o - - -2 0 1] and [1 1 −1 −1]. If X BC, find the (3, 1)-entry of X-¹ and the (2, 4)-entry of (XT)−¹. Explain your answers fully. (b) Let A and B be n × n matrices, and let y be a non-zero vector in R”. Show that if A - B is invertible, then the equations Ax=y and Bx = y do not have a common solution x ER". Hint: Suppose that x is some solution to both equations, and aim for a contradiction.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
I keep trying and I can't get the answer to this question. I would really appreciate if someone would help me. Thank you.
![(a) Suppose that B and C are invertible 4 × 4 matrices satisfying the
following:
• Columns 1 and 2 of B-¹ are, respectively,
2
-2
1.
-2
-
and
=
3
-2
.Rows 3 and 4 of C-1 are, respectively,
[o
-
-
-2 0 1] and [1 1 −1 −1].
If X
BC, find the (3, 1)-entry of X-¹ and the (2, 4)-entry of (XT)−¹.
Explain your answers fully.
(b) Let A and B be n × n matrices, and let y be a non-zero vector in R”.
Show that if A - B is invertible, then the equations
Ax=y and Bx = y
do not have a common solution x ER". Hint: Suppose that x is some
solution to both equations, and aim for a contradiction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e6ffc60-dbd1-49c1-89a5-60a078176da1%2F477b60b6-1ea3-4ac6-87af-62da3cd747c4%2F82le3w_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Suppose that B and C are invertible 4 × 4 matrices satisfying the
following:
• Columns 1 and 2 of B-¹ are, respectively,
2
-2
1.
-2
-
and
=
3
-2
.Rows 3 and 4 of C-1 are, respectively,
[o
-
-
-2 0 1] and [1 1 −1 −1].
If X
BC, find the (3, 1)-entry of X-¹ and the (2, 4)-entry of (XT)−¹.
Explain your answers fully.
(b) Let A and B be n × n matrices, and let y be a non-zero vector in R”.
Show that if A - B is invertible, then the equations
Ax=y and Bx = y
do not have a common solution x ER". Hint: Suppose that x is some
solution to both equations, and aim for a contradiction.
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