2.5 - Determinants & Multiplicative Inverses of Matrices.
-
Upload
horatio-adams -
Category
Documents
-
view
241 -
download
2
Transcript of 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