1. Downhill product – Uphill product.
Copy columns 1 & 2 to the outside C1 C2
Downhill products – Uphill products.

2. = (2)(5)(-9) + (-1)(1)(0) + (3)(-2)(6) – (0)(5)(3) – (6)(1)(2)
C C2
= (2)(5)(-9)
+ (-1)(1)(0)
+ (3)(-2)(6)
– (0)(5)(3)
– (6)(1)(2)
– (-9)(-2)(-1) =
+ (-36) – –
– (-18)
= -120
C C2
= (1)(10)(4)
+ (2)(10)(0)
+ (3)(2)(6)
– (0)(10)(3)
– (6)(10)(1)
– (4)(2)(2) = – – –
= 0

3. Uses determinants to solve systems of equations.
Matrix Dy is the coefficient matrix with the results in the y column..
Matrix D is the coefficient matrix.
Matrix Dx is the coefficient matrix with the results in the x column.

4. Set up all the determinants to solve the 3 by 3 system by Cramer’s Rule.
Find the coefficient matrix first.
Find Matrix Dx , Dy , and Dz for the numerators.

5. Solve the system by Cramer’s Rule.

6. Solve the system by Cramer’s Rule.
Remember to always find Matrix D 1st. If this matrix determinant is equal to zero, then there is no way to determine the answer. It is either NO SOLUTION, or INFINITE SOLUTION. We can’t tell.

7. Rows by Columns.
21 means Row 2 Column 1. If there are 2 digit rows or columns, a comma is used.
Equivalent Matrices must have the same dimensions and all corresponding entries must be the same.
Find the values of x, y, and z.
x = 4
y = - 8
z = 2

9. Zero Matrix is when every entry is zero.
Distributive Property with a Matrix..

10. Same
2 x 3
3 x 2
Same
The product is a 2 x 2 Matrix. Make an empty matrix.
2 ( 4 ) + 1 ( 8 ) + 3 ( 5 )
2 ( -7 ) + 1 ( 6 ) + 3 ( 9 )
1 ( 4 ) + -1 ( 8 ) + 0 ( 5 )
1 ( -7 ) + -1 ( 6 ) + 0 ( 9 )
1. Row 1 of A times Column 1 of B
2. Row 1 of A times Column 2 of B
3. Row 2 of A times Column 1 of B
4. Row 2 of A times Column 2 of B

11. Find BA
The product is a 3 x 3 Matrix. Make an empty matrix.
4 ( 2 ) + -7 ( 1 )
4 ( 1 ) + -7 ( -1 )
4 ( 3 ) + -7 ( 0 )
8 ( 2 ) + 6 ( 1 )
8 ( 1 ) + 6 ( -1 )
8 ( 3 ) + 6 ( 0 )
5 ( 2 ) + 9 ( 1 )
5 ( 1 ) + 9 ( -1 )
5 ( 3 ) + 9 ( 0 )
3 x 2
2 x 3
Same
1. R1 of B times C1 of A
2. R1 of B times C2 of A
3. R1 of B times C3 of A
4. R2 of B times C1 of A
5. R2 of B times C2 of A
6. R2 of B times C3 of A
7. R3 of B times C1 of A
8. R3 of B times C2 of A
9. R3 of B times C3 of A

12. The Identity Matrix is a square matrix, n by n matrix, which has 1’s on the downhill diagonal and 0’s for all the other entries.
Identity Matrix
Inverse Matrix.
Show that A and B are inverses.

13. One way is to create a variable matrix and multiply it to the matrix you are asked to find the inverse and set the product equal to the Identity Matrix. This creates systems of equations to solve.
How to find the inverse of a matrix?
2. R1 * C2
1. R1 * C1
4. R2 * C2
3. R2 * C1
Subtract the equations.
Back substitute.

14. Gauss-Jordan Method R1 – R2 = R1 -2R1 + R2 = R2
Here is another way to find the inverse of a matrix.
We can also make an Augmented Matrix, page 5 of the notes, and use the Gauss-Jordan Method to covert the matrix.
Gauss-Jordan Method -2 -0 -2 +2
R1 – R2 = R1
-2R1 + R2 = R2

15. Solving systems of equations using the inverse matrix method.
This technique is all calculator. We need to create a Linear Matrix Equation. x 6 1 1 -1 y 3 -2 1 -5 z 1 3 -2 14
We need to create a Coefficient Matrix …
Variable Matrix …
& Result Matrix.
Store Matrix A and B in calculator.
The Linear Matrix Equation.
Multiply both side by the inverse of Matrix A.
Find the inverse of Matrix A.
Determinant of A is the denominator in the inverse.

16. 3 ways to solve systems of equations
Gauss-Jordan Elimination Method, rref(
Cramer’s Rule, determinants
Inverse Matrix Method, inverse matrices
This is the best technique! The other two techniques can’t
determine when a system has No Solution or Infinite Solutions!