2.5 - Determinants & Multiplicative Inverses of Matrices.

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2.5 - Determinants & Multiplicative Inverses of Matrices

Transcript of 2.5 - Determinants & Multiplicative Inverses of Matrices.

Page 1: 2.5 - Determinants & Multiplicative Inverses of Matrices.

2.5 - Determinants & Multiplicative Inverses

of Matrices

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DETERMINANT

a real number representation of a square matrix.

The determinant of is a number denoted as or det

a matrix with a nonzero determinant is called nonsingular

a bc d

⎣⎢

⎦⎥

a bc d

a bc d

⎣⎢

⎦⎥

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Second-Order Determinant

The value of det or

is ad - cb.

a bc d

⎣⎢

⎦⎥

a bc d

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Examples1. Find the value

of

2. Find the value of

det 0 −28 −6

⎣⎢

⎦⎥

8 47 6

0(-6) - 8(-2)= 16

8(6) - 7(4)= 20

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page 98

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Third-Order Determinant

a1 b1 c1a2 b2 c2a3 b3 c3

=a1

b2 c2

b3 c3

−b1

a2 c2

a3 c3

+c1

a2 b2

a3 b3

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Find the value of 5 3 −16 4 80 −3 7

5 4 8−3 7

−3 6 80 7

+−1 6 40 −3

5 4(7)−−3(8)( )−3 6(7)−0(8)( )+−1 6(−3)−0(4)( )

=152

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Option 2 for finding

5 3 −16 4 80 −3 7

560

34−3

5 3 −16 4 80 −3 7

140 + 0 + 18 - 0 - -120 - 126

= 152

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The Identity Matrix

a square matrix whose elements in the main diagonal, from upper left to lower right, are 1s, while all other elements are 0s.

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Inverse Matrixthe product of a matrix and it’s inverse produces the identity matrix

only for square matrices

The inverse of matrix A would be denoted as A-1

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Inverse of a Second-Order Matrix

First, the matrix must be nonsingular!

Then, if the matrix is nonsingular, an inverse exists.

If the detA = 0, then it is singular and no inverse exists.

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Inverse of a Second-Order Matrix

If A = and ,

then A-1 =

a bc d

⎣⎢

⎦⎥

a bc d

≠0

1

a bc d

d −b−c a

⎣⎢

⎦⎥

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Find the inverse of8 93 −1

⎣⎢

⎦⎥

1st - find the det 8(-1) - 3(9) = -

35

2nd - find the inverse or−

1

35−1 −9−3 8

⎣⎢

⎦⎥

135

935

335 −8

35

⎢⎢⎢

⎥⎥⎥

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DAY 2

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Let’s use some technology!

it is important that you know how to do all these operations by hand.

matrices bigger than a second order are time consuming and well as multiplying matrices.

your calculators do all of this, but remember you will have a non-calculator section of your test.

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are solving systems and matrices in the same chapter?

You can use inverse matrices to

solve systems of linear equations!

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If we rewrite the system

as a product of matrices:

4x−2y=16x+6y=17

4 −21 6

⎣⎢

⎦⎥g

xy

⎣⎢⎢

⎦⎥⎥= 16

17⎡

⎣⎢

⎦⎥

Now, if this were a simple linear equation, like 5x = 15, how would

you “get rid of” the 5?

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First, find the inverse of

Then, multiply both sides by the inverse.

4 −21 6

⎣⎢

⎦⎥

xy

⎣⎢⎢

⎦⎥⎥=

313

113

−126

213

⎢⎢⎢

⎥⎥⎥g 16

17⎡

⎣⎢

⎦⎥

xy

⎣⎢⎢

⎦⎥⎥= 5

2⎡

⎣⎢

⎦⎥ (5, 2)

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Use inverse matrices to solve

5x + 4y=−3−3x−5y=−24