Definition 1 (Reduction Algorithm). Given a square matrix A, we can apply the reduction algorithm to obtain PALU where U is an upper triangular matrix, Lis a lower triangular matrix with 1's on the diagonal, and P is a permutation matrix. Then det (P) det(A) = ±det(A) det (L) det (U)= det(U) det (U) = U11. Unn. = Definition 2 (Cofactor Formula). det(A) where Cij = (-1)+det(Mij and Mij is the submatrix of A with row i and column j removed. Definition . Σ sgn(o)a10(1)... ano (n). σESym[n] n Σ aij Cij j=1 = 3 (Leibniz Formula). det(A)
Definition 1 (Reduction Algorithm). Given a square matrix A, we can apply the reduction algorithm to obtain PALU where U is an upper triangular matrix, Lis a lower triangular matrix with 1's on the diagonal, and P is a permutation matrix. Then det (P) det(A) = ±det(A) det (L) det (U)= det(U) det (U) = U11. Unn. = Definition 2 (Cofactor Formula). det(A) where Cij = (-1)+det(Mij and Mij is the submatrix of A with row i and column j removed. Definition . Σ sgn(o)a10(1)... ano (n). σESym[n] n Σ aij Cij j=1 = 3 (Leibniz Formula). det(A)
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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Question
![Definition 1 (Reduction Algorithm). Given a square
matrix A, we can apply the reduction algorithm to obtain
P. A = L U where U is an upper triangular matrix, L is
a lower triangular matrix with 1's on the diagonal, and P is
a permutation matrix. Then det (P) det(A) = ±det(A)
det (L) det (U)= det(U)
det (U) = U₁1. Unn.
=
Definition
2 (Cofactor Formula). det(A)
where Cij = (-1)+det(Mij and M₁, is the submatrix of A
with row i and column j removed.
Definition
.
Σ sgn(o)a10(1)... ano (n).
σESym[n]
n
Σ ajj Cij
j=1
=
3 (Leibniz Formula). det(A)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb9a7c18-65f1-48ed-bb7e-4937a04e4157%2F31e55610-468e-41c0-9ff9-b6c9c9e74800%2F70d9i3j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Definition 1 (Reduction Algorithm). Given a square
matrix A, we can apply the reduction algorithm to obtain
P. A = L U where U is an upper triangular matrix, L is
a lower triangular matrix with 1's on the diagonal, and P is
a permutation matrix. Then det (P) det(A) = ±det(A)
det (L) det (U)= det(U)
det (U) = U₁1. Unn.
=
Definition
2 (Cofactor Formula). det(A)
where Cij = (-1)+det(Mij and M₁, is the submatrix of A
with row i and column j removed.
Definition
.
Σ sgn(o)a10(1)... ano (n).
σESym[n]
n
Σ ajj Cij
j=1
=
3 (Leibniz Formula). det(A)

Transcribed Image Text:For each matrix below, compute the determinant using all three methods.
1 2 3
4 4
6 7
2 3
12
2 1
10
0 1
0 1 1
(a) A=4
5
1
3
3
(b) A =
1
(c) A = 1
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