(i) If A is n xn, then A and AT have the same eigenvalues. (ii) If A is nxn, then A and AT have the same eigenvectors. (iii) If A is n x n then det(Ak) = [det(A)]k (iv) If I is the nxn identity matrix, and J is an nxn matrix consisting entirely of ones, then the matrix I-is invertible and (1-1)¯¹ = 1+ J. n+I (v) If I is the nxn identity matrix, and J is an nxn matrix consisting entirely of ones, then the matrix A=I-is idempotent (i.e., A² = A). J n+I Don't forget that you are selecting which statements are false (you are not selecting which statements are true). (A) (ii) and (iv) (B) (i) and (v) (C) (ii) and (v) (D) (i) and (ii) (E) (iv) and (v) (F) (i) and (iv) (G) (iii) and (v)
(i) If A is n xn, then A and AT have the same eigenvalues. (ii) If A is nxn, then A and AT have the same eigenvectors. (iii) If A is n x n then det(Ak) = [det(A)]k (iv) If I is the nxn identity matrix, and J is an nxn matrix consisting entirely of ones, then the matrix I-is invertible and (1-1)¯¹ = 1+ J. n+I (v) If I is the nxn identity matrix, and J is an nxn matrix consisting entirely of ones, then the matrix A=I-is idempotent (i.e., A² = A). J n+I Don't forget that you are selecting which statements are false (you are not selecting which statements are true). (A) (ii) and (iv) (B) (i) and (v) (C) (ii) and (v) (D) (i) and (ii) (E) (iv) and (v) (F) (i) and (iv) (G) (iii) and (v)
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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![Problem #3: Consider the following 5 statements. 2 of the statements are false in general. Determine which 2 statements are
false by testing out each statement on an appropriate matrix (like we did with the properties of determinants in
Section 3.3 of the tutorial file).
Problem #3:
Note: You should not use a magic or pascal matrix for (i) or (ii) below because they have special properties
not shared by other matrices. Try using rand instead.
(i) If A is n xn, then A and AT have the same eigenvalues.
(ii) If A is nxn, then A and AT have the same eigenvectors.
(iii) If A is n x n then det(Ak ) = [det(A)]k
(iv) If I is the n× n identity matrix, and J is an n x n matrix consisting entirely of ones,
then the matrix I-is invertible and (1- ↓ )−¹ = I + J.
n+I
n+1
(v) If I is the n× n identity matrix, and J is an n x n matrix consisting entirely of ones,
then the matrix A=I-_ J
is idempotent (i.e., A² = A).
n+1
Don't forget that you are selecting which statements are false (you are not selecting which statements are true).
(A) (ii) and (iv) (B) (i) and (v) (C) (ii) and (v) (D) (i) and (ii) (E) (iv) and (v) (F) (i) and (iv) (G) (iii) and (v)
(H) (iii) and (iv)
Select](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324d2399-7c16-4255-9612-7ea336b734b4%2Fb071e349-3114-4505-b2ea-2ae0eb81576e%2Fg157fl_processed.png&w=3840&q=75)
Transcribed Image Text:Problem #3: Consider the following 5 statements. 2 of the statements are false in general. Determine which 2 statements are
false by testing out each statement on an appropriate matrix (like we did with the properties of determinants in
Section 3.3 of the tutorial file).
Problem #3:
Note: You should not use a magic or pascal matrix for (i) or (ii) below because they have special properties
not shared by other matrices. Try using rand instead.
(i) If A is n xn, then A and AT have the same eigenvalues.
(ii) If A is nxn, then A and AT have the same eigenvectors.
(iii) If A is n x n then det(Ak ) = [det(A)]k
(iv) If I is the n× n identity matrix, and J is an n x n matrix consisting entirely of ones,
then the matrix I-is invertible and (1- ↓ )−¹ = I + J.
n+I
n+1
(v) If I is the n× n identity matrix, and J is an n x n matrix consisting entirely of ones,
then the matrix A=I-_ J
is idempotent (i.e., A² = A).
n+1
Don't forget that you are selecting which statements are false (you are not selecting which statements are true).
(A) (ii) and (iv) (B) (i) and (v) (C) (ii) and (v) (D) (i) and (ii) (E) (iv) and (v) (F) (i) and (iv) (G) (iii) and (v)
(H) (iii) and (iv)
Select
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