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THU, JAN 8, 2015
Create a “Big Book of Matrices” flip book using 4 pages.
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BIG BOOK OF MATRICES
• What is a Matrix?
• Adding & Subtracting
• Multiply by a Scalar
• Multiplication
• Solve Systems using RREF
• Finding an Inverse Matrix
• Solve Systems using Inverses
MATRICES
What is a Matrix?
• A matrix is an array of numbers written in rows and columns.
• The dimension indicates how may rows and columns it has (rows first,
columns second).
• Matrix is singular, Matrices is plural
2x3
3 −2
0 4
1
5
3x2
−2 1
4
1
0 −1
3x1
1
−9
3
• A Square Matrix has the same number of Rows and Columns.
2x2
1 −8
2 −3
Adding & Subtracting
• In order to add or subtract matrices the dimension must be the same.
• If the dimensions are not the same…cannot be added or subtracted!
1
3 2
+
5
4 −1
−2
0
=
4 0
9 −1
−1 −1
0 8
1 4 − 3
5
0 −4
2 −2
=
1 2
+ 7 −2 =
4 −3
1 2
+ 7 −2
4 −3
NO SOLUTION
1
9
−2 −1
2
2
Scalar Multiplication
• A Scalar is just a number you are multiplying by, or distributing.
3 0
4
=
2 −1
4
1
−3 −1 + 2 3
2
5
12 0
8 −4
=
−12
2
3 + 6 =
−6
10
0 1
4
+ 3 3 −1 =
−3 2
−10
9
4
0
4
+ 9
−12 8
−3
Multiplication
• Only compatible matrices can be multiplied.
• Compatible: 2x3 • 3x1
2x1 • 1x6
3x2 • 2x2
3x3 • 3x3
• NOT Compatible: 2x3 • 2x3
2x2 • 3x2
1x2 • 3x1
2x3
• 3x1
1
2 1 3
∙ 2
0 2 4
3
2x2
•
=
2x1
Please see video under
Recommended Resources to
help you go through the
process of multiplication.
=
2x3
=
2x3
2 −2
0 1
2
∙
=
1 3
4 −1 −2
2x1 •
2
1
∙
4
2
2x2
= Not Compatible
3
=
5
2
1
∙
4
2
3
5
NO SOLUTION
Solving Systems using RREF
• RREF is Reduced Row Echelon Form
• AKA Gauss-Jordan Elimination
• Use for a system of Linear Equations!
Solve the system algebraically using Reduced Row Echelon Form:
x + y = -1
Please see video under
2x – 3y = 13
Recommended Resources to help you
solve systems using row reducing
Solve the system using the calculator:
•
•
•
•
•
Go to MATRIX, then EDIT.
Type in the dimension (this is a 2x3)
Type in the numbers
Go to MATRIX, then MATH
Find RREF (don’t forget to tell the calculator what matrix to RREF!)
Finding the Inverse of a Matrix
• Only Square Matrices can have Inverses
• Not ALL Square Matrices have Inverses, some are undefined (If the
Determinant = 0, NO INVERSE)
• If you have Matrix A, the Inverse of Matrix A is denoted by A-1
Please see video under
Recommended Resources to help you
find an inverse of a matrix.
Find the inverse:
2 1
7 4
Use your calculator to find the Inverse of each Matrix:
−1
A=
5
A-1 =
2
4
1
B = −2
5
B-1 =
2 3
3 1
4 0
C=
C-1 =
2 3
4 6
Solve Systems using Inverses
• The A Matrix contains the coefficients, the X Matrix contains the variables and the
B Matrix contains what the equations equal: AX = B
• When solving, you cannot divide by Matrix A, you MUST use the Inverse, so:
X = A-1B
A • X = B
X =
A-1 • B
3
1
𝑥
𝑥
𝑥
x + y = -1
−1
1 1
−1
2
5
5
=
∙
∙
=
=
2
−1
𝑦
𝑦
2x – 3y = 13
13
2 −3 𝑦
13
−3
5
5
Solve each system on the calculator using Inverse Matrices.
1. 3𝑥 − 4𝑦 = 17
2. 𝑥 + 𝑦 − 𝑧 = −3
2𝑥 + 𝑦 = −11
3𝑥 − 𝑦 + 4𝑧 = 0
−𝑥 − 2𝑦 + 𝑧 = 1
• Type in the A Matrix and the B Matrix
• Find A-1B