int math 2 section 8-5
DESCRIPTION
Matrices and DeterminantsTRANSCRIPT
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SECTION 8-5Matrices and Determinants
Tue, Apr 12
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ESSENTIAL QUESTIONS
How do you find the determinant of a 2X2 matrix?
How do you solve systems of equations using determinants?
Where you’ll see this:
Sports, construction, fitness
Tue, Apr 12
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VOCABULARY
1. Square Matrix:
2. Determinant:
3. Cramer’s Rule:
Tue, Apr 12
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VOCABULARY
1. Square Matrix: A matrix with the same number of rows and columns
2. Determinant:
3. Cramer’s Rule:
Tue, Apr 12
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VOCABULARY
1. Square Matrix: A matrix with the same number of rows and columns
2. Determinant: In a 2X2 matrix
a b
c d
⎡
⎣⎢
⎤
⎦⎥, det A = ad - bc.
3. Cramer’s Rule:
Tue, Apr 12
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VOCABULARY
1. Square Matrix: A matrix with the same number of rows and columns
2. Determinant: In a 2X2 matrix
a b
c d
⎡
⎣⎢
⎤
⎦⎥, det A = ad - bc.
3. Cramer’s Rule: A method of using determinants of matrices to solve systems of equations
Tue, Apr 12
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m x n: m rows and n columns
MATRIX
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m x n: m rows and n columns
MATRIX
−2 2 3
9 6 −3
⎡
⎣⎢
⎤
⎦⎥
3 −1
5 8
⎡
⎣⎢
⎤
⎦⎥
4 −3 0⎡⎣ ⎤⎦
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m x n: m rows and n columns
MATRIX
−2 2 3
9 6 −3
⎡
⎣⎢
⎤
⎦⎥
3 −1
5 8
⎡
⎣⎢
⎤
⎦⎥
4 −3 0⎡⎣ ⎤⎦
2 X 3
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m x n: m rows and n columns
MATRIX
−2 2 3
9 6 −3
⎡
⎣⎢
⎤
⎦⎥
3 −1
5 8
⎡
⎣⎢
⎤
⎦⎥
4 −3 0⎡⎣ ⎤⎦
2 X 3 2 X 2
Tue, Apr 12
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m x n: m rows and n columns
MATRIX
−2 2 3
9 6 −3
⎡
⎣⎢
⎤
⎦⎥
3 −1
5 8
⎡
⎣⎢
⎤
⎦⎥
4 −3 0⎡⎣ ⎤⎦
2 X 3 2 X 2 1 X 3
Tue, Apr 12
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DETERMINANT
det A =
a b
c d= ad − bc
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EXAMPLE 1
Find the determinant of
0 4
6 7
⎡
⎣⎢
⎤
⎦⎥.
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EXAMPLE 1
Find the determinant of
0 4
6 7
⎡
⎣⎢
⎤
⎦⎥.
ad − bc
Tue, Apr 12
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EXAMPLE 1
Find the determinant of
0 4
6 7
⎡
⎣⎢
⎤
⎦⎥.
ad − bc
= 0(7) − 4(6)
Tue, Apr 12
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EXAMPLE 1
Find the determinant of
0 4
6 7
⎡
⎣⎢
⎤
⎦⎥.
ad − bc
= 0(7) − 4(6)
= 0 − 24
Tue, Apr 12
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EXAMPLE 1
Find the determinant of
0 4
6 7
⎡
⎣⎢
⎤
⎦⎥.
ad − bc
= 0(7) − 4(6)
= 0 − 24
= −24
Tue, Apr 12
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CRAMER’S RULE
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CRAMER’S RULE
1. Make sure equations look like Ax + By = C.
Tue, Apr 12
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CRAMER’S RULE
1. Make sure equations look like Ax + By = C.
2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A.
Tue, Apr 12
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CRAMER’S RULE
1. Make sure equations look like Ax + By = C.
2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A.
3. Make a new 2X2 determinant matrix: Replace x column with equation answers, call Ax.
Tue, Apr 12
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CRAMER’S RULE
1. Make sure equations look like Ax + By = C.
2. Make a 2X2 determinant matrix: x in 1st column, y in 2nd, call A.
3. Make a new 2X2 determinant matrix: Replace x column with equation answers, call Ax.
4. Make another 2X2 determinant matrix: Replace y column with equation answers, call Ay.
Tue, Apr 12
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CRAMER’S RULE
Tue, Apr 12
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CRAMER’S RULE
5. Divide Ax by A and Ay by A.
Tue, Apr 12
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CRAMER’S RULE
5. Divide Ax by A and Ay by A.
6. Check answer and rewrite solution.
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
=−12 + 77
6 + 7
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
=−12 + 77
6 + 7 =
6513
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
=−12 + 77
6 + 7 =
6513 = 5
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
=−12 + 77
6 + 7 =
6513 = 5
y =
Ay
A
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
=−12 + 77
6 + 7 =
6513 = 5
y =
Ay
A =
(3)(11) − (−6)(1)(3)(2) − (−7)(1)
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
=−12 + 77
6 + 7 =
6513 = 5
y =
Ay
A =
(3)(11) − (−6)(1)(3)(2) − (−7)(1)
=33 + 66 + 7
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
=−12 + 77
6 + 7 =
6513 = 5
y =
Ay
A =
(3)(11) − (−6)(1)(3)(2) − (−7)(1)
=33 + 66 + 7
=3913
Tue, Apr 12
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EXAMPLE 2Solve the system of equations using Cramer’s rule (matrices).
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
A =3 −7
1 2 Ax =
−6 −7
11 2 Ay =
3 −6
1 11
x =
Ax
A =
(−6)(2) − (−7)(11)(3)(2) − (−7)(1)
=−12 + 77
6 + 7 =
6513 = 5
y =
Ay
A =
(3)(11) − (−6)(1)(3)(2) − (−7)(1)
=33 + 66 + 7
=3913 = 3
Tue, Apr 12
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EXAMPLE 2
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
x = 5, y = 3
Tue, Apr 12
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EXAMPLE 2
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
x = 5, y = 3
Check:
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EXAMPLE 2
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
x = 5, y = 3
Check: 3(5) − 7(3) = −6
Tue, Apr 12
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EXAMPLE 2
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
x = 5, y = 3
Check: 3(5) − 7(3) = −6
15 − 21 = −6
Tue, Apr 12
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EXAMPLE 2
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
x = 5, y = 3
Check: 3(5) − 7(3) = −6
15 − 21 = −6 5 + 2(3) =11
Tue, Apr 12
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EXAMPLE 2
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
x = 5, y = 3
Check: 3(5) − 7(3) = −6
15 − 21 = −6 5 + 2(3) =11
5 + 6 =11
Tue, Apr 12
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EXAMPLE 2
3x − 7 y = −6
x + 2 y =11
⎧⎨⎩
x = 5, y = 3
Check: 3(5) − 7(3) = −6
15 − 21 = −6 5 + 2(3) =11
5 + 6 =11
(5,3)
Tue, Apr 12
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PROBLEM SET
Tue, Apr 12
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PROBLEM SET
p. 356 #1-31 odd
Solve all using matrices by hand
“I’m a great believer in luck, and I find the harder I work the more I have of it.” - Thomas Jefferson
Tue, Apr 12