Let i be an eigenvalue of an invertible matrix A. Show that 1 is an eigenvalue of A1. [Hint: Suppose a nonzero x satisfies Ax = Ax.] Note that A' exists. In order for A1 to be an eigenvalue of A', there must exist 1 nonzero x such that A -1x=-1x. Suppose a nonzero x satisfies Ax = Ax. What is the first operation that should be performed on Ax = Ax so that an equation similar to the one in the previous step can be obtained? O A. Right-multiply both sides of Ax = Ax by A1. O B. Invert the product on each side of the equation. 1 O C. Left-multiply both sides of Ax = ix by A Perform the operation and simplify. (Type an equation. Simplify your answer.) Why does this show that A1 is defined? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. Since the product 2 'x must be defined and nonzero, A1 must exist and be nonzero. O B. Since x is an eigenvector of A, A1 and x are commutable. By definition, x is nonzero, so the previous equation cannot be satisfied if A = O C. By definition, x is nonzero and A is invertible. So, the previous equation cannot be satisfied if A= 0. How does this show that 11 is an eigenvalue of A1? Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer.) O A. Both sides of the equation can be multiplied by and one side can be simplified to obtain 1x =Ax. O B. Both sides of the equation can be multiplied by and one side can be simplified to obtain (A-A-11)x = 0. O C. Both sides of the equation can be multiplied by and one side can be simplified to obtain 21A-x = x. O D. Both sides of the equation can be multiplied by and one side can be simplified to obtain 21A-x = 0.
Let i be an eigenvalue of an invertible matrix A. Show that 1 is an eigenvalue of A1. [Hint: Suppose a nonzero x satisfies Ax = Ax.] Note that A' exists. In order for A1 to be an eigenvalue of A', there must exist 1 nonzero x such that A -1x=-1x. Suppose a nonzero x satisfies Ax = Ax. What is the first operation that should be performed on Ax = Ax so that an equation similar to the one in the previous step can be obtained? O A. Right-multiply both sides of Ax = Ax by A1. O B. Invert the product on each side of the equation. 1 O C. Left-multiply both sides of Ax = ix by A Perform the operation and simplify. (Type an equation. Simplify your answer.) Why does this show that A1 is defined? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. Since the product 2 'x must be defined and nonzero, A1 must exist and be nonzero. O B. Since x is an eigenvector of A, A1 and x are commutable. By definition, x is nonzero, so the previous equation cannot be satisfied if A = O C. By definition, x is nonzero and A is invertible. So, the previous equation cannot be satisfied if A= 0. How does this show that 11 is an eigenvalue of A1? Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer.) O A. Both sides of the equation can be multiplied by and one side can be simplified to obtain 1x =Ax. O B. Both sides of the equation can be multiplied by and one side can be simplified to obtain (A-A-11)x = 0. O C. Both sides of the equation can be multiplied by and one side can be simplified to obtain 21A-x = x. O D. Both sides of the equation can be multiplied by and one side can be simplified to obtain 21A-x = 0.
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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![Let i be an eigenvalue of an invertible matrix A. Show that A is an eigenvalue of A'. [Hint: Suppose a nonzero x satisfies Ax = Ax.]
-1x=-1x.
1
Note that A' exists. In order for A1 to be an eigenvalue of A', there must exist
nonzero x such that A
Suppose a nonzero x satisfies Ax = Ax. What is the first operation that should be performed on Ax = Ax so that an equation similar to the one in the previous step can be obtained?
O A. Right-multiply both sides of Ax = Ax by A1.
O B. Invert the product on each side of the equation.
1
O C. Left-multiply both sides of Ax = 1x by A
Perform the operation and simplify.
(Type an equation. Simplify your answer.)
Why does this show that A1 is defined? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. Since the product
'x must be defined and nonzero, A must exist and be nonzero.
O B. Since x is an eigenvector of A, A
and x are commutable. By definition, x is nonzero, so the previous equation cannot be satisfied if A =
O C. By definition, x is nonzero and A is invertible. So, the previous equation cannot be satisfied if = 0.
How does this show that 1 is an eigenvalue of A? Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
O A. Both sides of the equation can be multiplied by and one side can be simplified to obtain 1x=Ax.
O B. Both sides of the equation can be multiplied by
and one side can be simplified to obtain (A--1)x = 0.
O C. Both sides of the equation can be multiplied by
and one side can be simplified to obtain 2A-x = x.
O D. Both sides of the equation can be multiplied by
and one side can be simplified to obtain 2A-x = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e6b254a-d25d-479c-b42b-752b8b5bbcbb%2F3aeacb29-990b-46e9-affa-3fb8bf54757e%2Fpkxrv9_processed.png&w=3840&q=75)
Transcribed Image Text:Let i be an eigenvalue of an invertible matrix A. Show that A is an eigenvalue of A'. [Hint: Suppose a nonzero x satisfies Ax = Ax.]
-1x=-1x.
1
Note that A' exists. In order for A1 to be an eigenvalue of A', there must exist
nonzero x such that A
Suppose a nonzero x satisfies Ax = Ax. What is the first operation that should be performed on Ax = Ax so that an equation similar to the one in the previous step can be obtained?
O A. Right-multiply both sides of Ax = Ax by A1.
O B. Invert the product on each side of the equation.
1
O C. Left-multiply both sides of Ax = 1x by A
Perform the operation and simplify.
(Type an equation. Simplify your answer.)
Why does this show that A1 is defined? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. Since the product
'x must be defined and nonzero, A must exist and be nonzero.
O B. Since x is an eigenvector of A, A
and x are commutable. By definition, x is nonzero, so the previous equation cannot be satisfied if A =
O C. By definition, x is nonzero and A is invertible. So, the previous equation cannot be satisfied if = 0.
How does this show that 1 is an eigenvalue of A? Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
O A. Both sides of the equation can be multiplied by and one side can be simplified to obtain 1x=Ax.
O B. Both sides of the equation can be multiplied by
and one side can be simplified to obtain (A--1)x = 0.
O C. Both sides of the equation can be multiplied by
and one side can be simplified to obtain 2A-x = x.
O D. Both sides of the equation can be multiplied by
and one side can be simplified to obtain 2A-x = 0.
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