# ECON 1150 Matrix Operations Special Matrices

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

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

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

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

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

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

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

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

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

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

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

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

Scalar multiplication Conformability: Not required
ECON 1150, Spring 2013

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

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

2x3 3x1 2x1 ECON 1150, Spring 2013

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

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

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

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

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

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