matrix algebra ib1
Post on 09-Apr-2018
230 Views
Preview:
TRANSCRIPT
-
8/7/2019 Matrix algebra ib1
1/29
MATRIX ALGEBRAUnit 4 IB SL Math
-
8/7/2019 Matrix algebra ib1
2/29
MATRIX: A rectangular
arrangement of numbers in
rows and columns.
The ORDER of a matrix isthe number of the rows and
columns.
The ENTRIES are the
numbers in the matrix.
-
502
126
rows
columns
This order of this matrixThis order of this matrix
is a 2 x 3.is a 2 x 3.
-
8/7/2019 Matrix algebra ib1
3/29
-
67237
89511
36402
-
3410
200
318 ? A0759
-
20
11
-
6
0
7
9
3 x 3
3 x 5
2 x 2 4 x 1
1 x 4
(or square
matrix)
(Also called a
row matrix)
(or square
matrix)
(Also called a
column matrix)
-
8/7/2019 Matrix algebra ib1
4/29
To add two matrices, they must have the sameorder. To add, you simply add correspondingentries.
-
-
34
03
12
70
43
35
-
!
)3(740
0433
13)2(5
-
!
44
40
23
-
8/7/2019 Matrix algebra ib1
5/29
-
9245
3108
-
2335
2571
? A)1(8 70 51 23
55 34 32 )2(9 =
= ?7 7 4 50 7 5 7 A
-
8/7/2019 Matrix algebra ib1
6/29
To subtract two matrices, they must have theTo subtract two matrices, they must have thesame order. You simply subtract correspondingsame order. You simply subtract correspondingentries.entries.
-
-
232
451704
831
605429
-
!
2833)2(1
)4(65015
740249
-
!
603
1054
325
-
8/7/2019 Matrix algebra ib1
7/29
-
-
724
113
810
051
708
342
? ?A=5-2
-4-1 3-8
8-3 0-(-1) -7-1
1-(-4)
2-0
0-7
=
2 -5 -5
5 1 -8
5 3 -7A
-
8/7/2019 Matrix algebra ib1
8/29
In matrix algebra, a real number is often called aSCALAR. To multiply a matrix by a scalar, you multiplyeach entry in the matrix by that scalar.
-
14
024
-
!
416
08
-
!
)1(4)4(4
)0(4)2(4
-
8/7/2019 Matrix algebra ib1
9/29
-
-
86
54
30
212
-
!
)8(360
52412
!
? ?? A
AA-2
6
-3 3
-2(-3)-5
!!
-2(6) -2(-5)
-2(3) 6 -6
-12 10
-
8/7/2019 Matrix algebra ib1
10/29
Equal MatricesTwo matrices are equal if the entries in corresponding positions
are equal
-
{
-
-
!
-
10
47
10
74
05.
23
0
2
123
-
8/7/2019 Matrix algebra ib1
11/29
1-11
MULTIPLICATIONOF MATRICES
-
8/7/2019 Matrix algebra ib1
12/29
1-12
-
8/7/2019 Matrix algebra ib1
13/29
1-13
EXAMPLE 5 CONTINUED
-
8/7/2019 Matrix algebra ib1
14/29
1-14
EXAMPLE 5 CONTINUED
-
8/7/2019 Matrix algebra ib1
15/29
NOTE
CW
For 2x2 (two by two) matrices A and B then AB { BA
-
8/7/2019 Matrix algebra ib1
16/29
45
23
Notice the different symbol:
the straight lines tell you to
find the determinant!!
(3 * 4) - (-5 * 2)
12 - (-10)
22
=
45
23
Finding Determinants of Matrices
=
=
-
8/7/2019 Matrix algebra ib1
17/29
241
212
21
-1
-2
4
= [(2)(-2)(2) + ( )( )(-1) + ( )(1)(4)]
[( )(-2)(-1) + (2)( )(4) + ( )(1)(2)]
[-8 + +12]
-
- [6 + 40 + 0]
4 6- 40
Finding Determinants of Matrices
=
= = -42
-
8/7/2019 Matrix algebra ib1
18/29
-
10
01
Identity matrix: Square matrix with 1s on the diagonal and zeros
everywhere else
2 x 2 identity matrix
-
100
010
001
3 x 3 identity matrix
The identity matrix is to matrix multiplication as ___ is to regularmultiplication!!!!1
Using matrix equations
-
8/7/2019 Matrix algebra ib1
19/29
-
10
01
Identity matrix: Square matrix with 1s on the diagonal and zeros
everywhere else
2 x 2 identity matrix
-
100
010
001
3 x 3 identity matrix
The identity matrix is to matrix multiplication as ___ is to regularmultiplication!!!!1
Using matrix equations
-
8/7/2019 Matrix algebra ib1
20/29
Multiply:
-
10
01
-
43
25=
-
43
25
-
1001
-
4325
=
-
4325
So, the identity matrix multiplied by any matrix
lets the any matrix keep its identity!
Mathematically, IA = A and AI = A !!
-
8/7/2019 Matrix algebra ib1
21/29
InverseMatrix:
Using matrix equations
2 x 2
-
dc
ba
In words:
Take the original matrix.Switch a and d.
Change the signs of b and c.
Multiply the new matrix by 1 over the determinant of the original
-
ac
bd
bcad
1!1A
!A
-
8/7/2019 Matrix algebra ib1
22/29
-
24
410
)4)(4()10)(2(
1
-
24
410
4
1=
-
2
11
125
Using matrix equations
Example: Find the inverse of A.
-
104
42!A
!1A
!1
A
-
8/7/2019 Matrix algebra ib1
23/29
Find the inverse matrix.
-
25
38
Det A =8(2) (-5)(-3) = 16 15= 1
Matrix A
Inverse =
-
det
1Matrix
Reloaded
-
85
321
1= =
-
8532
-
8/7/2019 Matrix algebra ib1
24/29
What happens when you multiply a matrix by its inverse?
1st: What happens when you multiply a number by its inverse?717 y
A & B are inverses. Multiply them.
-
85
32=
-
25
38
-
10
01
So, AA-1 = I
-
8/7/2019 Matrix algebra ib1
25/29
Why do we need to know all this? To Solve Problems!
Solve for Matrix X.
=
-
2538 X
-
1314
We need to undo the coefficient matrix. Multiply it by its INVERSE!
-
8532 =
-
2538 X
-
8532
-
1314
-
10
01
X=
-
34
11
X =
-
34
11
-
8/7/2019 Matrix algebra ib1
26/29
You can take a system of equations and write it with
matrices!!!
3x + 2y = 11
2x + y = 8becomes
-
12
23
-
y
x=
-
8
11
Coefficient
matrix
Variable
matrix
Answer matrix
Using matrix equations
-
8/7/2019 Matrix algebra ib1
27/29
Let A be the coefficient matrix.
Multiply both sides of the equation by the inverse of A.
-
!
-
-
!
-
-
!
-
8
11
8
11
8
11
1
11
Ay
x
Ay
xAA
y
xA
-
12
23 -1=
-
32
21
1
1=
-
32
21
-
32
21
-
12
23
-
y
x=
-
32
21
-
8
11
-
1001
-yx =
-
25
-
y
x=
-
2
5
Using matrix equations
-
12
23
-
y
x=
-
8
11Example: Solve for x and y .
!1A
-
8/7/2019 Matrix algebra ib1
28/29
Wow!!!!
3x + 2y = 11
2x + y = 8
x = 5; y = -2
3(5) + 2(-2) = 11
2(5) + (-2) = 8
It works!!!!
Using matrix equations
Check:
-
8/7/2019 Matrix algebra ib1
29/29
You Try
Solve:
4x + 6y = 14
2x 5y = -9
(1/2,2)
top related