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**ECON 1150 Matrix Operations Special Matrices**

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

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**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

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**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

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**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

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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

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**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

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**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

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**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

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**2. Matrices and Vectors = A matrix is a rectangular array of numbers.**

ECON 1150, Spring 2013

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**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

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**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

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**3. Matrix Operations Addition and subtraction**

Adding up elements of the same corresponding position Conformability: Same order Example 6.1: ECON 1150, Spring 2013

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**Scalar multiplication Conformability: Not required**

ECON 1150, Spring 2013

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**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

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

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2x3 3x1 2x1 ECON 1150, Spring 2013

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**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

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**ECON 1150, Spring 2013 ECON 1150 Spring 2013**

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

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**Remark: (AB)C = A(BC) A(B + C) = AB + AC (A + B)C = AC + BC**

ECON 1150, Spring 2013

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**4. Special Matrices 4.1 Square Matrices # of rows = # of columns E.g.,**

ECON 1150, Spring 2013

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**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

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**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

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