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Techniques of ROW REDUCTION can simplify the process of evaluating the determinant of a matrix. Replace row 2 with row 2 + k(row 1). - PowerPoint PPT Presentation
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Techniques of
ROW REDUCTION
can simplify the process of
evaluating the determinant of a matrix
db
ca
db
ca
Replace row 2 with row 2 + k(row 1)
kcdkab
ca
db
ca
Replace row 2 with row 2 + k(row 1)
bcaddb
ca
det
kcdkab
ca
db
ca
Replace row 2 with row 2 + k(row 1)
bcaddb
ca
det
kcdkab
ca
bcad
akcbcakcad
kabckcda
kcdkab
ca
)()(
det
db
ca
Replace row 2 with row 2 + k(row 1)
bcaddb
ca
det
kcdkab
ca
bcad
akcbcakcad
kabckcda
kcdkab
ca
)()(
det
When you replace a row of a matrix with itself plus a multiple of another row, the determinant does not change.
542
372
371
det
542
001
371
det
542
372
371
det
Replace row 2 with row 2 – row 1
23)1235)(1(54
37det)1(
542
001
371
det
542
372
371
det
db
ca
db
ca
Interchange row 1 and row 2
ca
db
db
ca
Interchange row 1 and row 2
bcaddb
ca
det
ca
db
db
ca
Replace row 2 with row 2 + k(row 1)
bcaddb
ca
det
)(
det
bcadadbc
ca
db
ca
db
db
ca
Replace row 2 with row 2 + k(row 1)
bcaddb
ca
det
)(
det
bcadadbc
ca
db
ca
db
When you interchange 2 rows of a matrix, the determinant changes by a factor of –1. This would not simplify the process of evaluating a determinant.
db
ca
db
ca
Replace row 1 with k(row 1)
db
kcka
db
cabcad
db
ca
det
Replace row 1 with k(row 1)
db
kcka
db
cabcad
db
ca
det
)(
det
bcadkkbckad
db
kcka
Replace row 1 with k(row 1)
db
kcka
db
cabcad
db
ca
det
When you replace a row of a matrix with k times itself, the determinant changes by a factor of k. This would not simplify the process of evaluating a determinant.
)(
det
bcadkkbckad
db
kcka
Replace row 1 with k(row 1)
db
kcka
example:
572
8103
241
det
910
220
241
1 row (2)times 3 row with 3 row replace
572
220
241
1 row (-3)times 2 row with 2 row replace
572
8103
241
910
220
241
det
1 row (2)times 3 row with 3 row replace
572
220
241
det
1 row (-3)times 2 row with 2 row replace
572
8103
241
det
2091
22det)1(
910
220
241
det
1 row (2)times 3 row with 3 row replace
572
220
241
det
1 row (-3)times 2 row with 2 row replace
572
8103
241
det
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