1. 8.1 Matrices & Systems of Equations
To solve a system of equations
Write system of equations as an augmented matrix
Use row operations to convert to row echelon form
Find variables using back-substitution
WE’LL DO EXAMPLES IN CLASS!
Augmented Matrix
Row Operations
Row Echelon Form –> Gaussian Elimination
Reduced Row Echelon Form -> Gauss-Jordan Elimination
Augmented Matrix Example 25 = 2z + 3y x 19 y 31 3x 25 2 3 1 19 31
Augmented Matrix in Row Echelon Form (Different Example) 4 = z 9 3z + y
-19 5z – 2y x 4 1 9 3
-19 -5 2
* Reduced Row Echelon Form has 1’s along the diagonal and 0’s everywhere else

2. Matrix Row Operations Two rows of a matrix may be interchanged.
The elements in any row may be multiplied by a nonzero number. (3R2)
Multiply a row by a non-zero number and add to another row
(Example: 2R1 + R3 -> R3) -6 4 -3 -2 5 2 1 21
-12 18 3
Perform row operations:
Interchange: R1   R2 3R1 2R2 + R3R3 -6 4 -3 -2 21
-12 18 3 5 2 1 -6 4 -3 -2 5 2 1 63
-36 54 9 4 -2 1 5 -3 2 21
-12 18 3

3. 8.2 Matrix Algebra A matrix of order m x n has m rows and n columns
A = [aij] (Matrix a with elements aij)
Order: 2 x 3
a23 = -1/5
a12. = 2
Matrix Addition é - ù é - ù é 2 1 3 2 1 3 = + ê ú ê ú ê 5 - - 4 - 5 1 - 3 - 1 ë 3 6 ë û ë û
Scalar Multiplication 3 é9 3 6 ù 3 = ê ú ë - 12 - 15 - 3 / 5 û
**Note: A matrix where all elements are zero is known as the zero matrix : 0

4. Equality of Matrices = =
Two matrices A and B are equal (A = B) if they have the same
Order (number of rows and columns are equal) and each (i,j) entry
Of A is equal to the corresponding (i,j) entry of B.
x – 8 = 10  x = 18
Y + 7 = 15  y = 8
V = 6 and R = 5
X – Y V R =
x + y = 4
X – y = 1
So, x = 5/2 and y = 3/2
X + Y
x - y =

5. Matrix Multiplication
Note: AB is often not equal to BA
Try A = X B =

6. 8.3 The Matrix Inverse X = = X = =
For the Real Number System: A x 1 = A and 1 x A = A
So, 1 is the multiplicative Identity
For Matrices: A x I = A and I x A = A
So, I is the multiplicative Identity Matrix.
I is a square matrix (2x2, 3x3, etc.) with 1’s in the Diagonal and 0’s elsewhere X = = X
For the Real Number System (A) (1/A) = 1 and (1/A)(A) = 1 (Identity)
So, 1/A is the multiplicative inverse of A and A is the multiplicative inverse of 1/A
For Matrices: AA-1 = I and A-1A = I (If AB = BA = I then A & B are inverses) = =

7. Finding a Matrix Inverse or Proving None
To prove a matrix has no inverse, suppose
has an inverse
Then,
Since the two matricies are equal y ou
Must have 0 = 1, but this is False,
Thus, No Inverse.
Find the multiplicative inverse of
w x
y z
2w + y = 1 2x + z = 0
5w + 3y = 0 5x + 3z = 1
Solve the system of equations
X = -1 w = 3 So, A-1 =
Y = -5 z = =
Special rule for inverse of a 2 x 2 matrix
Let A= a b If ad – bc =0 the no inverse
c d Otherwise, A-1 =

8. Another Method for Finding an Inverse
Start With:
Perform Row Operations until
The left is the Identity Matrix
Thus, the inverse is:

9. Solving Systems of Equations using Inverse
To solve the system, solve
The matrix equation : AX = B
 X = A-1B
Write the linear system in matrix form, then find the inverse. We know the inverse of this matrix since we found it in the previous example.
A X = B
So, the solution set is {(3, 1, 4)}

10. 8.4 Determinants and Cramer’s Rule
The determinant of a 2 x 2 matrix, A is denoted det(A), |A| or
|A| = ad - bc
Example:
The determinant of a square n x n matrix, A, (n ≥ 3).
is the sum of the entries in any row of A (or column of A), multiplied by their respective cofactors.
The minor Mij of the element aij is the determinant of the (n–1) × (n–1) matrix obtained by deleting the ith row and the jth column of A.
The cofactor Aij of the entry aij is given by:

11. Finding Minors and Cofactors
a. Find minors M11 and M32
b. Find cofactors A11 and A32
c. Find |A|
To find M11 delete the first row and first column
Then, find the determinant:
(-3)(7) – (-6)(1) = = = -15
To find M32 delete the third row and second column
Then, find the determinant:
(-6)(2) – (4)(5) = = -32
To find cofactors A11 & A32
A11 = (-1)1+1 (-15) = (-1)2 (-15) = 1(15) = 15
A32 = (-1)3+2 (-32) = (-1)5 (-32) = -1(-32) = -32

12. Example: Find the Determinant
Recall: The determinant of a square n x n matrix,
is the sum of the entries in any row (or column),
multiplied by their respective cofactors.

13. CRAMER’S RULE FOR SOLVING TWO EQUATIONS IN TWO VARIABLES
The system
of two equations in
two variables has a unique solution (x, y) given by
provided that D ≠ 0, where
© 2010 Pearson Education, Inc. All rights reserved

14. © 2010 Pearson Education, Inc. All rights reserved
Using Cramer’s Rule to Solve Systems of Equations
Solve the system (2 x 2):
Solve the system(3 x 3)
Recall:
|A| = a11A11+a21A21+a31A31
© 2010 Pearson Education, Inc. All rights reserved