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Lesson 8Determinants and Inverses (Section 13.5–6)
Math 20
October 5, 2007
Announcements
I No class Monday 10/8, yes class Friday 10/12
I Problem Set 3 is on the course web site. Due October 10
I Sign up for conference times on course website
I Prob. Sess.: Sundays 6–7 (SC 221), Tuesdays 1–2 (SC 116)
I OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
Review: Determinants of n × n matrices by patterns
DefinitionLet A = (aij)n×n be a matrix. The determinant of A is a sum ofall products of n elements of the matrix, where each product takesexactly one entry from each row and column.
The sign of each product is given by (−1)σ, where σ is the numberof upwards lines used when all the entries in a pattern areconnected.
Review: Determinants of n × n matrices by patterns
DefinitionLet A = (aij)n×n be a matrix. The determinant of A is a sum ofall products of n elements of the matrix, where each product takesexactly one entry from each row and column.The sign of each product is given by (−1)σ, where σ is the numberof upwards lines used when all the entries in a pattern areconnected.
4× 4 sudoku patterns
+ − − + + −
− + + − − +
+ − − + + −
− + + − − +
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Determinants of n × n matrices by cofactors
DefinitionLet A = (aij)n×n be a matrix. The (i , j)-minor of A is the matrixobtained from A by deleting the ith row and j column. This matrixhas dimensions (n − 1)× (n − 1).The (i , j) cofactor of A is the determinant of the (i , j) minortimes (−1)i+j .
Math 20 - October 05, 2007.GWBFriday, Oct 5, 2007
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FactThe determinant of A = (aij)n×n is the sum
a11C11 + a12C12 + · · ·+ a1nC1n
FactThe determinant of A = (aij)n×n is the sum
a11Ci1 + ai2Ci2 + · · ·+ ainCin
for any i .
FactThe determinant of A = (aij)n×n is the sum
a1jC1j + a2jC2j + · · ·+ anjCnj
for any j.
Example
Compute the determinant:
∣∣∣∣∣∣2 −4 33 1 21 4 −1
∣∣∣∣∣∣I Expand along 1st row
I Expand along 2nd row
I Expand along 1st column
Math 20 - October 05, 2007.GWBFriday, Oct 5, 2007
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FactThe determinant of A = (aij)n×n is the sum
a11C11 + a12C12 + · · ·+ a1nC1n
FactThe determinant of A = (aij)n×n is the sum
a11Ci1 + ai2Ci2 + · · ·+ ainCin
for any i .
FactThe determinant of A = (aij)n×n is the sum
a1jC1j + a2jC2j + · · ·+ anjCnj
for any j.
FactThe determinant of A = (aij)n×n is the sum
a11C11 + a12C12 + · · ·+ a1nC1n
FactThe determinant of A = (aij)n×n is the sum
a11Ci1 + ai2Ci2 + · · ·+ ainCin
for any i .
FactThe determinant of A = (aij)n×n is the sum
a1jC1j + a2jC2j + · · ·+ anjCnj
for any j.
Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| = |A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| =
0.
2. |A′| = |A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| = |A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| =
|A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
Math 20 - October 05, 2007.GWBFriday, Oct 5, 2007
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Math 20 - October 05, 2007.GWBFriday, Oct 5, 2007
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Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| = |A|
3. If B is the matrix obtained by multiplying each entry of onerow or column of A by the same number α, then |B| = α |A|.
4. If two rows or columns of A are interchanged, then thedeterminant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
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Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| = |A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.
4. If two rows or columns of A are interchanged, then thedeterminant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| = |A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.4. If two rows or columns of A are interchanged,
then thedeterminant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
Math 20 - October 05, 2007.GWBFriday, Oct 5, 2007
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Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| = |A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| = |A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| =
0.
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Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A′| = |A|3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.
Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| = 0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| = αn |A|.
Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| =
0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| = αn |A|.
Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| = 0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| = αn |A|.
Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| = 0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then
the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| = αn |A|.
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Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| = 0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| = αn |A|.
Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| = 0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| = αn |A|.
Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| = 0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then
|αA| = αn |A|.
Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| = 0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| =
αn |A|.
Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then|A| = 0.
6. If a scalar multiple of one row (or column) of A is added toanother row (or column), then the determinant does notchange.
7. The determinant of the product of two matrices is the productof the determinants of those matrices:
|AB| = |A| |B|
8. if α is any real number, then |αA| = αn |A|.
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