4.1: Matrix OperationsObjectives: Students will be able to:• Add, subtract, and multiply a matrix by a scalar• Solve Matrix Equations• Use matrices to organize data
MatrixA rectangular arrangement of numbers in
rows and columns
Dimensions of a Matrix: # rows by # columns
2 X 3 (read 2 by 3)
701
436
Entries: the numbers in a matrix
Square Matrix: a matrix with the same # of rows and columns
What are the dimensions of the matrices below?
1. 2. 3.
3241
1071
265305712
Two matrices are equal if their dimensions are the same and the entries in corresponding positions are equal.
Are the following matrices equal?
1. and 2. and
441
217
425.
5.7
5321
3810
To add and subtract matrices, add or subtract corresponding entries:
Can only add and subtract if matrices have the same dimensions
Perform the indicated operations:1. 2. 3.
891
340
53
025933
8431
79
Scalar Multiplication: multiply each entry of the matrix by the scalar
1.
2.
4832
6
418963
923374
1
Solve for x and y:
71959
2410
52
3y
x
71959
2410
31563y
x
71959
231953y
x
3x = -9, x = -33y-2 =7, y = 3
Properties of Matrix Operations: A, B and C are matrices, c is a scalar
1. Associative Property (regroup)
2. Commutative Property (change order
3. Distributive Property of Addition
4. Distributive Property of Subtractions
(A+B)+C = A +(B+C)
A + B = B +A
c(A +B) = cA + cB
c(A- B) = cA- cB
Using Matrices to Organize Data:
Use matrices to organize the following data about insurance rates.
This year for 1 car, comprehensive, collision and basic insurance cost $612.15, $518.29 and $486.91. For 2 cars, comprehensive, collision and basic insurance cost $1150.32, $984.16, and $892.51.Next year for 1 car, comprehensive, collision and basic insurance will cost $616.28, $520.39, and $490.05. For 2 cars, comprehensive, collision and basic insurance will cost $1155.84, $987.72, and $895.13.
Use the matrices to write a matrix that shows the changes from this year to next.
51.89291.48616.98429.51832.115015.612
1 car 2 carsComp.
Coll.
basic
13.89505.49072.98739.52084.115528.616
This year (A) Next year (B)
B – A will give the change of:
62.214.356.31.252.513.4
Multiplying Matrices
-You can only multiply matrices when the number of columns in the first matrix is equal to the number of rows in the second.
-Multiplication of matrices is not commutative!!
-The dimensions of the product matrix will be the number of rows in the first matrix by the number of columns in the second matrix
3 x 2 matrix times a 2 x 2 matrix results in a 3 x 2 matrix
Multiply each row entry by each column entry to yield one entry in the product matrix.
831020211320
312
1 x 3 3 x 3
Must be the same
Dimensions of product matrix
DeterminantsThe determinant of a matrix is the
difference in the cross products
det A or lAl
bcdadcba
Find the determinant of a 2 x 2 matrix
14410)14()52(5412
Using Diagonals• Another method for evaluating a third
order determinant is using diagonals. • STEP 1: You begin by repeating the first
two columns on the right side of the determinant.
heb
gda
ihgfedcba
ihgfedcba
Using Diagonals• STEP 2: Draw a diagonal from each element in
the top row diagonally downward. Find the product of the numbers on each diagonal.
heb
gda
ihgfedcba
ihgfedcba
aei bfg cdh
Using Diagonals• STEP 3: Then draw a diagonal from each
element in the bottom row diagonally upward. Find the product of the numbers on each .
heb
gda
ihgfedcba
ihgfedcba
idbhfagec
Using Diagonals• To find the value of the determinant, add
the products in the first set of diagonals, and then subtract the products from the second set of diagonals.
The value is: aei + bfg + cdh – gec – hfa – idb
Ex. 2: Evaluate using diagonals.
124
331
213523041
213523041
First, rewrite the first two columns along side the determinant.
Ex. 2: Evaluate using diagonals.
124
331
213523041
213523041
Next, find the values using the diagonals.
4 60 0
0 -5 24
Now add the bottom products and subtract the top products.
4 + 60 + 0 – 0 – (-5) – 24 = 45. The value of the determinant is 45.
Area of a triangle• Determinants can be used to find the
area of a triangle when you know the coordinates of the three vertices. The area of a triangle whose vertices have coordinates (a, b), (c, d), (e, f) can be found by using the formula:
,111
21
fedcba
A and then finding |A|, since the area cannot be negative.
Ex. 3: Find the area of the triangle whose vertices have coordinates (-4, -1), (3, 2), (4, 6).
How to start: Assign values to a, b, c, d, e, and f and substitute them into the area formula and evaluate.
,111
21
fedcba
A a = -4, b = -1, c = 3, d = 2, e = 4, f = 6
621
434
164123114
21
A
-8 -4 18
8 -24-3
Now add the bottom products and subtract the top products.
-8 + (-4) + 18 – 8 – (-24) –(-3) = 25. The value of the determinant is 25. Applied to the area formula ½ (25) = 12.5. The area of the triangle is 12.5 square units.
1) Inverse Matrices and Systems of Equations
For a We can write a System of Equations Matrix Equation
145352
yxyx
145
5321yx
1) Inverse Matrices and Systems of Equations
Example 1:Write the system as a matrix equation
Matrix Equation
621132
yxyx
611
2132yx
Coefficient matrix
Constant matrix
Variable matrix
1) Inverse Matrices and Systems of Equations
BAX 1
When rearranging, take the inverse of A
BAX
1) Inverse Matrices and Systems of Equations
Example 3:Solve the system
Step 3: Solve for the variable matrix
14
611
2132
1
1
yx
yx
BAyx
BAX
The solution to the system is (4, 1).
621132
yxyx
1) Inverse Matrices and Systems of Equations
Example 2:
A BX
850
212121111
zyx
822520
zyxzyxzyx
Solving systems using Augmented Matrices
• You can solve some linear systems by using an augmented matrix. An augmented matrix contains the coefficients and the constants from a system of equations. Each row of the matrix represents an equation.
Ex. 1: Write an augmented matrix to represent the system shown.
2041026
xyx
System of Equations -6x - 2y = 10
4x = -20
System of Equations
2010
|04|26
Use the rref key under the matrix math menu to solve an augmented matrix.
Ex. 2: Write an augmented matrix to represent the system shown.
333155
yxyx
System of Equations x - 5y = 15
3x +3y = 3
System of Equations
315|33|51
An augmented matrix that represents the system
Ex. 3: Write an augmented matrix to represent the system shown.
382
432
zzy
zyx
System of Equations x + 2y +3z = -4
y – 2z = 8
z = -3
System of Equations
310082104321
An augmented matrix that represents the system