1. ECON 1150 Matrix Operations Special Matrices
Spring 2013
Lecture 6
Linear Algebra
Matrix Operations
Special Matrices
ECON 1150, Spring 2013

2. 1. Systems of Linear Equations
ECON 1150
Spring 2013
1. Systems of Linear Equations
Consider the following equations:
x1 + 2x2 = (1)
2x1 – x2 = (2)
Method of substitution
Method of elimination
Solution:
(x1*, x2*)
A solution of this system is a pair of numbers (x1*, x2*) which satisfies both equations.
The solution of a system of equations remains the same after row operations.
R2 – 2*R1
x1 + 2x2 = (3)
- 5x2 = (4)
R4 / (-5)
x1 + 2x2 = (5)
x2 = (6)
R5 – 2*R6
x1 = (7)
x2 = (8)
ECON 1150, Spring 2013

3. Consider the following equations: 4x1 – x2 + 2x3 = 13 (1)
ECON 1150
Spring 2013
Consider the following equations:
4x1 – x2 + 2x3 = (1)
x1 + 2x2 – 2x3 = (2)
-x1 + x2 + x3 = (3)
How is the solution (x1*, x2*, x3*) found?
ECON 1150, Spring 2013

4. Multiply an equation by a nonzero constant
ECON 1150
Spring 2013
Row operations:
Multiply an equation by a nonzero constant
Add a multiple of one equation to another
Interchange any two equations
Gaussian Elimination
The solution of a system of equations remains the same after row operations.
This method can be applied to any number of equations.
Step 1: Switch (R1) and (R2),
x x2 – 2x3 = (R4)
4x1 – x x3 = (R5)
–x x x3 = (R6)
Step 2: (R5) + 4(R4),
x x2 – 2x3 = (R7)
– 9x x3 = (R8)
–x x x3 = (R9)
Step 3: (R9) + (R7),
x x2 – 2x3 = (R10)
– 9x x3 = (R11)
3x2 – x3 = (R12)
Step 4: (R11) / (-9),
x x2 – 2x = (R13)
x2 – (10/9)x3 = –13/ (R14)
3x2 – x3 = (R15)
Step 5: (R15) – 3(R14),
x x2 – 2x = (R16)
x2 – (10/9)x3 = –13/ (R17)
(7/3)x3 = 28/ (R18)
Step 6: (R18) * 3 / 7,
x x2 – 2x = (R19)
x2 – (10/9)x3 = –13/ (R20)
x3 = (R21)
Step 7: (R20) + 10 / 9 * (R21)
x x2 – 2x3 = (R22)
x = (R23)
x3 = (R24)
Step 8: (R22) – 2(R23) + 2(R24),
x1* = 2, x2* = 3, x3* = 4
x1* = 2, x2* = 3, x3* = 4
ECON 1150, Spring 2013

5. ECON 1150
Spring 2013
General rule: To solve a system of linear equations with a unique solution, the number of linearly independent equations must be equal to the number of variables.
If any equation of a system of equations can be derived from a series of row operations on that system, then this system is called linearly dependent. Otherwise, it is called linearly independent.
ECON 1150, Spring 2013

6. The following systems are linearly dependent.
ECON 1150
Spring 2013
The following systems are linearly dependent.
2x + y = (1)
x – y = (2)
-1.5x + 3y = (3)
– x – y z = – (4)
3x + 2y – 2z = (5)
x – y z = (6)
0.5 (R1) – 2.5 (R2) = (R3)
5(R4) + 2(R5) ) = (R6)
ECON 1150, Spring 2013

7. m linearly independent equations n unknowns
ECON 1150
Spring 2013
Equation system:
m linearly independent equations
n unknowns
Case 1: m = n  unique solution
x + y = 10, x – y = 6
Case 2: m < n  Infinitely many solutions
x + y = 10
Case 3: m > n  No solution.
x + y = 10, x – y = 4, 2x – 3y = 6
ECON 1150, Spring 2013

8. General equation system
Matrix algebra can help to
simplify the expression
solve the system efficiently
testing the existence of a solution
ECON 1150, Spring 2013

9. 2. Matrices and Vectors = A matrix is a rectangular array of numbers.
ECON 1150, Spring 2013

10. Order (dimension) of a matrix = (# of rows) x (# of columns)
Row vector: a matrix with only one row
Column vector: a matrix with only one column
[ ]
ECON 1150, Spring 2013

11. Two matrices are equal if they have the same order and the corresponding elements are equal. E.g.,
2x2
x = y = 2.
ECON 1150, Spring 2013

12. 3. Matrix Operations Addition and subtraction
Adding up elements of the same corresponding position
Conformability: Same order
Example 6.1:
ECON 1150, Spring 2013

13. Scalar multiplication Conformability: Not required
ECON 1150, Spring 2013

14. Rule of matrix multiplication:
ECON 1150
Spring 2013
Multiplication
Conformability: The number of the columns of A is equal to the number of rows of B.
Rule of matrix multiplication:
The element of the ith row and jth column of matrix C
ith row of A  jth column of B
and adding the resulting products.
Write down all 3 matrices on the board.
ECON 1150, Spring 2013

15. 1x3
3x1
1x1
1x3
3x2
1x2
ECON 1150, Spring 2013

16. 2x3
3x1
2x1
ECON 1150, Spring 2013

17. Let C = AB = . Then c1 = 4(7) + 11(2) c2 = 17(7) + 6(2)
ECON 1150
Spring 2013
Let C = AB = Then
c1 = 4(7) + 11(2)
c2 = 17(7) + 6(2)
Note that BA is not well defined.
ECON 1150, Spring 2013

18. ECON 1150, Spring 2013 ECON 1150 Spring 2013
Note that BA is a 3 by 3 matrix.
ECON 1150, Spring 2013

19. Remark: (AB)C = A(BC) A(B + C) = AB + AC (A + B)C = AC + BC
ECON 1150, Spring 2013

20. 4. Special Matrices 4.1 Square Matrices # of rows = # of columns E.g.,
ECON 1150, Spring 2013

21. A matrix whose elements are all zero.
ECON 1150
Spring 2013
4.2 Identity Matrices
For any M by N matrix A,
IMA = AIN = A.
An identity matrix in matrix operation is like the number “1” in real number.
A null matrix is like zero in real numbers.
4.3 Null Matrices
A matrix whose elements are all zero.
ECON 1150, Spring 2013

22. A square matrix that is equal to its transpose.
4.4 The transpose of a matrix A is a new matrix AT such that the ith row of A is the ith column of AT.
4.5 Symmetric matrices
A square matrix that is equal to its transpose.
ECON 1150, Spring 2013