4. If X is a square matrix, and X has partition where A is an invertible submatrix of X, then where A = X = det (X) = det (A) det(X/A). Here, X/A = D-CA-¹B is the Schur complement of A in X. Use this principle to find the determinant of (²3) X A B C D A B) ₁ CD = 12 1 3 23 2 5 13 1 2 (3 7 - −1 1
4. If X is a square matrix, and X has partition where A is an invertible submatrix of X, then where A = X = det (X) = det (A) det(X/A). Here, X/A = D-CA-¹B is the Schur complement of A in X. Use this principle to find the determinant of (²3) X A B C D A B) ₁ CD = 12 1 3 23 2 5 13 1 2 (3 7 - −1 1
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.5: Subspaces, Basis, Dimension, And Rank
Problem 63EQ
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Here, this question (attached) covers the concept of Shur Complement. Given question is attached, along with my attempt. For some reason my professor marked my answer wrong. Could anyone help provide some clarity on why I may have gotten my answer wrong? I was not given any prior feedback regarding this question.
Transcribed Image Text:4. If X is a square matrix, and X has partition
where A is an invertible submatrix of X, then
where A =
X =
det(X) = det (A) det(X/A).
Here, X/A = D-CA-¹B is the Schur complement of A in X. Use this principle to find the
determinant of
12
23
X =
A B).
CD
A
(& B)
=
12 1 3
23 2 5
13 1 2
(3 7 −1 1
![A B
4. Xisq matrix end x has partition x- (CO). A i invertible submatoril
of X, then dol (x) = del (A) det (X/A). Here, X/A = D-CA-¹B is the
shur complement of Ain X. Use this principle to find the detominent
of x = (CB) = (1 ²
23 2 5
13.12
37-11
13
12
-A= (23), B-(25) C = (₁ ²³₁), 0= (17₂)
37
- dex (A)-+-1), A* (21)
• X/A-O-CA-1 B
13)-(33) (23) (33
4-1
O
C
12
-11
)(
X/A = (33)-(33 340) = (-32-32)
-11.
d-bc • det (x/A) = [(-12) + (-19)]-[(-32) × (-32)]
det (x/A)
=-76
35
513
12
where A = (23
13
113
25
• det (x) = det (A) det(x/A)
bet (A) dot (VA)
det(x) = -1 × 76
76
X](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99690567-bdca-4632-b99f-493e44b366e1%2F4b2be151-bf22-467d-9ee7-d1915be9f011%2Fpzmwf9o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A B
4. Xisq matrix end x has partition x- (CO). A i invertible submatoril
of X, then dol (x) = del (A) det (X/A). Here, X/A = D-CA-¹B is the
shur complement of Ain X. Use this principle to find the detominent
of x = (CB) = (1 ²
23 2 5
13.12
37-11
13
12
-A= (23), B-(25) C = (₁ ²³₁), 0= (17₂)
37
- dex (A)-+-1), A* (21)
• X/A-O-CA-1 B
13)-(33) (23) (33
4-1
O
C
12
-11
)(
X/A = (33)-(33 340) = (-32-32)
-11.
d-bc • det (x/A) = [(-12) + (-19)]-[(-32) × (-32)]
det (x/A)
=-76
35
513
12
where A = (23
13
113
25
• det (x) = det (A) det(x/A)
bet (A) dot (VA)
det(x) = -1 × 76
76
X
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