2.5 - Determinants & Multiplicative Inverses of Matrices

2.5 - Determinants & Multiplicative Inverses

of Matrices

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

⎡

⎣⎢

⎤

⎦⎥

Second-Order Determinant

The value of det or

is ad - cb.

a bc d

⎡

⎣⎢

⎤

⎦⎥

a bc d

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

page 98

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

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

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

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.

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

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.

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

⎡

⎣⎢

⎤

⎦⎥

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

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

DAY 2

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.

are solving systems and matrices in the same chapter?

You can use inverse matrices to

solve systems of linear equations!

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?

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)

Use inverse matrices to solve

5x + 4y=−3−3x−5y=−24