Techniques of

ROW REDUCTION

can simplify the process of

evaluating the determinant of a matrix

dbca

dbca

Replace row 2 with row 2 + k(row 1)

kcdkabca

dbca

Replace row 2 with row 2 + k(row 1)

bcaddbca

det

kcdkabca

dbca

Replace row 2 with row 2 + k(row 1)

bcaddbca

det

kcdkabca

bcadakcbcakcad

kabckcdakcdkab

ca

)()(

det

dbca

Replace row 2 with row 2 + k(row 1)

bcaddbca

det

kcdkabca

bcadakcbcakcad

kabckcdakcdkab

ca

)()(

det

When you replace a row of a matrix with itself plus a multiple of another row, the determinant does not change.

542372371

det

542001371

det542372371

det

Replace row 2 with row 2 – row 1

23)1235)(1(5437

det)1(542001371

det542372371

det

dbca

dbca

Interchange row 1 and row 2

cadb

dbca

Interchange row 1 and row 2

bcaddbca

det

cadb

dbca

Replace row 2 with row 2 + k(row 1)

bcaddbca

det

)(

det

bcadadbccadb

cadb

dbca

Replace row 2 with row 2 + k(row 1)

bcaddbca

det

)(

det

bcadadbccadb

cadb

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.

dbca

dbca

Replace row 1 with k(row 1)

dbkcka

dbca

bcaddbca

det

Replace row 1 with k(row 1)

dbkcka

dbca

bcaddbca

det

)(

det

bcadkkbckaddbkcka

Replace row 1 with k(row 1)

dbkcka

dbca

bcaddbca

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

bcadkkbckaddbkcka

Replace row 1 with k(row 1)

dbkcka

example:

5728103241

det

910220241

1 row (2)times 3 row with 3 row replace

572220241

1 row (-3)times 2 row with 2 row replace

5728103241

910220241

det

1 row (2)times 3 row with 3 row replace

572220241

det

1 row (-3)times 2 row with 2 row replace

5728103241

det

209122

det)1(910220241

det

1 row (2)times 3 row with 3 row replace

572220241

det

1 row (-3)times 2 row with 2 row replace

5728103241

det