Use cofactor expansion to show that the determinant of an upper-triangular matrix is the product of its diagonal elements. Remark: Using Gaussian elimination, one could thus compute det(A) in O(n³) steps, - rather than in O(n!) steps that would be required for the brute-force formula (1). Namely we have the following Lemma: Let A E Rnxn. Apply elementary row operations {E₁}=1 in order to obtain B and define 1 di = a -1 if E, is an eliminate operation if E, is a scale-by-a operation if E, is an exchange operation. Then det (B) = det (A) II di. Proof: By Lemma 1, det is linear in each row of the matrix and negated by a row exchange. By linearity, det is not affected by an eliminate operation.
Use cofactor expansion to show that the determinant of an upper-triangular matrix is the product of its diagonal elements. Remark: Using Gaussian elimination, one could thus compute det(A) in O(n³) steps, - rather than in O(n!) steps that would be required for the brute-force formula (1). Namely we have the following Lemma: Let A E Rnxn. Apply elementary row operations {E₁}=1 in order to obtain B and define 1 di = a -1 if E, is an eliminate operation if E, is a scale-by-a operation if E, is an exchange operation. Then det (B) = det (A) II di. Proof: By Lemma 1, det is linear in each row of the matrix and negated by a row exchange. By linearity, det is not affected by an eliminate operation.
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:9. Use cofactor expansion to show that the determinant of an upper-triangular matrix is the
product of its diagonal elements.
Remark: Using Gaussian elimination, one could thus compute det(A) in O(n³) steps, rather than in
O(n!) steps that would be required for the brute-force formula (1). Namely we have the following Lemma:
Let A E Rnxn. Apply elementary row operations {E₁}=1 in order to obtain B and define
di:
=
1
a
-1
if E, is an eliminate operation
if E, is a scale-by-a operation
if E is an exchange operation.
-
=
Then det (B) = det (A) 1 d. Proof: By Lemma 1, det is linear in each row of the matrix and negated
by a row exchange. By linearity, det is not affected by an eliminate operation.
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