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Matrices: What Are They?
10th Grade Algebra II Honors Project
by
TweetMegan Dempsey
on 27 May 2014Transcript of Matrices: What Are They?
Scalar Multiplication
Addition and Subtraction
Transpose of a Matrix
Matrix Multiplication
Matrix Operations
Megan Caroline Dempsey  Honors
Matrices: What Are They?
What Are Matrices?
3
2
6
5
2
7
10
8
3
Rectangular array of numbers
Named by dimensions; ex. 2x3, 3x3, 2x4, etc.
Way of representing information:
Computer graphics/science (numbers represent other things, ex. color intensity, objects, movement. etc.)
Physics (quantum mechanics, electrical circuits, optics, etc.)
Stochastic matrices and Eigen vector solvers used in ranking of Google search.
7
5
10
3
8
0
3
x
Scalar
=
21
15
30
9
24
0
1 7 5
0 3 10
+
5 0 3
11 1 7
=
6 7 8
11 2 3
0 1
3 2

1 3
0 5
=
1
2
3
3
Different Dimensions
3 2 0
4 6 7
+
5 7
1 6
2 x 2
2 x 3
=
Undefined
Cannot + or  matrices of different dimensions, just like you cannot add, subtract, multiply or divide roots of different #'s.
3
√
2xyz
√
3xyz
1 2
3 0
7 5
A =
A
T
=
transpose of A
Transpose = rows become columns
columns become rows
A
T
=
1 3 7
2 0 5
2
2
5
3
X
1
4
7
6
2(1) + 2 (7)
2(4) + 2(6)
5(1) + 2(7)
5(4) + 3(6)
16 20
9 2
2 x 3 times 3 x 2
column
row
Order matters: ED ≠ DE, i.e. Matrix multiplication is not commutative
Properties of Matrix Multiplication
Is Associative: A(BC) = (AB)C
Identity Matrix
1 x a = a
1 x b = b
1 x 5 = 5
1 = identity for scalar numbers
2 X 2 Matrix:
1 0
0 1
3 x 3 Matrix:
1 0 0
0 1 0
0 0 1
4 X 4 Matrix:
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Inverting a 2x2 Matrix
finding the determinant
Inverse 2 x 2:
1
determinant
x
Adjugate
Example: Matrix A
A =
5 3
1 4
determinant =
5 3
1 4
5(4)  3(1)
Determinant of Matrix A =
23
Adjugate
Example Matrix A:
Matrix A =
3 5
7 2
Adjugate =
3 5
7 2
()
()
Adjugate of A =
2 5
7 3
3 5
7 2
Inverse of:
determinant = 41
1
41
x
2 5
7 3
Inverse =
2
41
5
41
7
41
3
41
A x A
1
=
Identity Matrix
Systems of Equations
3x + 2y = 7
6x + 6y = 6
=
3 2
6 6
x
y
=
7
6
A
b
c
Ab = c
multiply everything by the inverse of A
A
1
Ab
=
A
1
c
Ib =
A
1
c
b =
A
1
c
Inverting a 3x3 Matrix
Finding the determinant
Method 1:
Example Matrix A:
4 1 1
4 5 3
2 0 0
4 1 1
4 5 3
2 0 0
4
4
2
1
5
0
+

4(5)(0) + 1(3)(2) + 1(4)(0)
1(4)(0)  4(3)(0)  1(5)(2)
0
6
0
0
0
10
10 + 6 = 16
Method 2:
4 1 1
4 5 3
2 0 0
4 1 1
4 5 3
2 0 0
+  +
+ 4
5 3
0 0

1
4 3
2 0
+
1
4 5
2 0
0
6
10
10 + 6 = 16
Example Matrix C=
1 2 2
2 1 1
3 4 5
Matrix of Minors
1 2 2
2 1 1
3 4 5
=
1 1
4 5
2 1
3 5
2 1
3 4
2 2
4 5
1 2
3 5
1  2
3 4
2 2
1 1
1 2
2 1
1 2
2 1
1 7 5
18 11 2
4 5 3
Cofactor Matrix
+  +
 + 
+  +
1 7 5
18 11 2
4 5 3
1 7 5
18 11 2
4 5 3
Determinant: 23
1
23
Cofactor matrix
T
1
23
1 18 4
7 11 5
5 2 3
1
23
18
23
4
23
7
23
11
23
5
23
5
23
2
23
3
23
Augmented Matrix
2x  3y = 8
4x + 5y = 1
2
4
3
5
8
1
Reduced Row Echelon
Used to find solutions for systems and tell whether it is dependent or inconsistent.
Rules:
Interchange rows
Multiply entries by a nonzero #
Add constant times row to another.
2
4
3
5
8
1
(1/2)
1
4
1.5
5
4
1
4
(+)
1
0
1.5
11
4
15
(1/11)
1
0
1.5
1
4
15/11
(3/2)
(+)
1
0
0
1
43/22
15/11
Full transcriptAddition and Subtraction
Transpose of a Matrix
Matrix Multiplication
Matrix Operations
Megan Caroline Dempsey  Honors
Matrices: What Are They?
What Are Matrices?
3
2
6
5
2
7
10
8
3
Rectangular array of numbers
Named by dimensions; ex. 2x3, 3x3, 2x4, etc.
Way of representing information:
Computer graphics/science (numbers represent other things, ex. color intensity, objects, movement. etc.)
Physics (quantum mechanics, electrical circuits, optics, etc.)
Stochastic matrices and Eigen vector solvers used in ranking of Google search.
7
5
10
3
8
0
3
x
Scalar
=
21
15
30
9
24
0
1 7 5
0 3 10
+
5 0 3
11 1 7
=
6 7 8
11 2 3
0 1
3 2

1 3
0 5
=
1
2
3
3
Different Dimensions
3 2 0
4 6 7
+
5 7
1 6
2 x 2
2 x 3
=
Undefined
Cannot + or  matrices of different dimensions, just like you cannot add, subtract, multiply or divide roots of different #'s.
3
√
2xyz
√
3xyz
1 2
3 0
7 5
A =
A
T
=
transpose of A
Transpose = rows become columns
columns become rows
A
T
=
1 3 7
2 0 5
2
2
5
3
X
1
4
7
6
2(1) + 2 (7)
2(4) + 2(6)
5(1) + 2(7)
5(4) + 3(6)
16 20
9 2
2 x 3 times 3 x 2
column
row
Order matters: ED ≠ DE, i.e. Matrix multiplication is not commutative
Properties of Matrix Multiplication
Is Associative: A(BC) = (AB)C
Identity Matrix
1 x a = a
1 x b = b
1 x 5 = 5
1 = identity for scalar numbers
2 X 2 Matrix:
1 0
0 1
3 x 3 Matrix:
1 0 0
0 1 0
0 0 1
4 X 4 Matrix:
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Inverting a 2x2 Matrix
finding the determinant
Inverse 2 x 2:
1
determinant
x
Adjugate
Example: Matrix A
A =
5 3
1 4
determinant =
5 3
1 4
5(4)  3(1)
Determinant of Matrix A =
23
Adjugate
Example Matrix A:
Matrix A =
3 5
7 2
Adjugate =
3 5
7 2
()
()
Adjugate of A =
2 5
7 3
3 5
7 2
Inverse of:
determinant = 41
1
41
x
2 5
7 3
Inverse =
2
41
5
41
7
41
3
41
A x A
1
=
Identity Matrix
Systems of Equations
3x + 2y = 7
6x + 6y = 6
=
3 2
6 6
x
y
=
7
6
A
b
c
Ab = c
multiply everything by the inverse of A
A
1
Ab
=
A
1
c
Ib =
A
1
c
b =
A
1
c
Inverting a 3x3 Matrix
Finding the determinant
Method 1:
Example Matrix A:
4 1 1
4 5 3
2 0 0
4 1 1
4 5 3
2 0 0
4
4
2
1
5
0
+

4(5)(0) + 1(3)(2) + 1(4)(0)
1(4)(0)  4(3)(0)  1(5)(2)
0
6
0
0
0
10
10 + 6 = 16
Method 2:
4 1 1
4 5 3
2 0 0
4 1 1
4 5 3
2 0 0
+  +
+ 4
5 3
0 0

1
4 3
2 0
+
1
4 5
2 0
0
6
10
10 + 6 = 16
Example Matrix C=
1 2 2
2 1 1
3 4 5
Matrix of Minors
1 2 2
2 1 1
3 4 5
=
1 1
4 5
2 1
3 5
2 1
3 4
2 2
4 5
1 2
3 5
1  2
3 4
2 2
1 1
1 2
2 1
1 2
2 1
1 7 5
18 11 2
4 5 3
Cofactor Matrix
+  +
 + 
+  +
1 7 5
18 11 2
4 5 3
1 7 5
18 11 2
4 5 3
Determinant: 23
1
23
Cofactor matrix
T
1
23
1 18 4
7 11 5
5 2 3
1
23
18
23
4
23
7
23
11
23
5
23
5
23
2
23
3
23
Augmented Matrix
2x  3y = 8
4x + 5y = 1
2
4
3
5
8
1
Reduced Row Echelon
Used to find solutions for systems and tell whether it is dependent or inconsistent.
Rules:
Interchange rows
Multiply entries by a nonzero #
Add constant times row to another.
2
4
3
5
8
1
(1/2)
1
4
1.5
5
4
1
4
(+)
1
0
1.5
11
4
15
(1/11)
1
0
1.5
1
4
15/11
(3/2)
(+)
1
0
0
1
43/22
15/11