Topic 4: Matrices Reading: Jacques: Chapter 7, Section

Transcription

1 Topic 4: Matrices Reading: Jacques: Chapter 7, Section dding, sutracting and multiplying matrices 2. Matrix inversion 3. pplication: National Income Determination

2 What is a matrix? Matrix is a two-dimensional array of numers arranged in rows and columns Convenient way of descriing data Example: Suppose a firm produces 3 goods and sells those goods to 2 consumers. Sales in June are given y: Sold to consumer June Sales G1 G2 G3 C C matrix is a convenient way of representing this information:

3 What is a matrix? Each entry is said to e an element of the matrix matrix that has m rows and n columns is said to e a matrix of order m x n They are denoted y capital letters in old type (e.g., B, C, ) Elements are denoted y their corresponding lower-case letter in ordinary type (e.g. a,, c, ) Suscript tells us which row (i) and column (j) the element enters (e.g. a ij, ij, c ij,.) Example: The matrix is of order 2x3: a11 a12 a a a a

4 What is a matrix? row vector is a matrix with only one row. Example: [ ] column vector is a matrix with only one column. Example: Basic Matrix Operations: Transposition 2 d 5 9 ddition and Sutraction Scalar Multiplication Matrix Multiplication

5 Transposition The transpose of a matrix replaces the rows with the columns The matrix is of order 2x3: a11 a12 a13 a21 a22 a 23 The transpose of is of order 3x2: a11 a21 T a12 a 22 a13 a 23

6 Transposition Example: Say we want to present the information as follows: Sold to consumer June Sales G1 G2 G3 C C Sold to Consumer C1 C2 June Sales G1 7 1 G2 3 5 G T

7 dding and Sutracting Matrices Matrices can e added or sutracted y adding and sutracting the corresponding elements Only matrices of the same order (same numer of rows and columns) can e added and sutracted

8 dding and Sutracting Matrices Example: Suppose a firm produces 3 goods and sells those goods to 2 consumers. Sales in June are given y: Sold to consumer Sales in July are given y: June Sales G1 G2 G3 C C Sold to consumer July Sales G1 G2 G3 C C What are total sales for June and July?

9 dding and Sutracting Matrices Write as matrices: B B C C is a matrix representing comined sales for June and July y good and consumer

10 dding and Sutracting Matrices Example: D E F 5 8 D 2 1 d ij e ij f ij 6 10 E 1 4 Note: D E F If and B are mxn matrices, then +BB+ If is an mxn matrix then 0 where 0 is a zero matrix of order mxn. Example:

11 Scalar Multiplication To multiply a matrix y a scalar k we multiply each element of y k: Example: k a k a k a k k a k a k a k a k a k a Suppose sales are the same every month as they are in June. What are total sales for the year y product and consumer? Examples

12 Matrix Multiplication To multiply a matrix y a matrix B, the numer of columns in must e equal to the numer of rows in B First consider the multiplication of 2 vectors a and 11 a [ a11 a12... a1s ] ( 1 s)... s 1 11 a s s s1... s1 4 Example: Find a a [ 1 2 3] s ( 1 ) 21 [ a a a ] a + a a ( 1 1) 11

13 Matrix Multiplication Example: Find a a [ 1 2 3] First check that they conform! a [ 1 2 3] ( 1 3) ( ) 4 a [ ] 5 1( 4) + 2( 5) + 3( 6) 32 ( 1 1) scalar 6 Example

14 Matrix Multiplication In general if is m x s and B is s x n then and B can e multiplied: x B C (m x s) (s x n) (m x n) c ij is found y multiplying the ith row of into the jth column of B Example: a [ a a a ] a x B c (1 x 3) (3 x 2) (1 x 2) 13, B c [ c11 c12 ] c11 a a a1331 Note: Order of new matrix

15 Matrix Multiplication In general if is m x s and B is s x n then and B can e multiplied: x B C (m x s) (s x n) (m x n) c ij is found y multiplying the ith row of into the jth column of B Example: a [ a a a ] a x B c (1 x 3) (3 x 2) (1 x 2) 13, B c11 a a a1331 c12 a a a c [ c11 c12 ]

16 Matrix Multiplication Example: Find cab a [ 4 ] 1 6, B c c Rememer: lways check order to make sure multiplication can e performed Example: Find CB ab c [ 28 35] , B

17 Matrix Properties - Summary + B B k( + B) k + kb k(l) (kl) (B + C) B +C Scalar Counterpart a++a a-a0 a+0a k(a+)ka+k k(la)(kl)a a(+c)a+ac ( + B) C C + BC (a+)cac+c (BC) (B)C a(c)(a)c B B aa

18 System of equations in Matrix form Matrix notation can e used to illustrate familiar mathematics prolems e.g. a system of linear equations: 4x + 3y 2x + 11 y 5 Can e expressed in Matrix notation as : where 4 2 3, 1 x x x, y 11 5 contains coefficients x contains unknowns contains right hand sides

19 Matrix Inversion Square matrix: numer of rows and columns are equal Identity matrix: analogous to numer 1 in ordinary arithmetic 2x2 Identity Matrix: 3x3 Identity Matrix: I I 1 0 I I Scalar Counterpart: 1.aa

20 Matrix Inversion Square matrix: numer of rows and columns are equal Identity matrix: analogous to numer 1 in ordinary arithmetic If 2x2: I I There is another matrix: a11 a12 a a21 a 22 c d 1 1 d ad c c a Scalar Counterpart: 1.aa 1 0 I 0 1 Such that: -1 1 I and I -1 is the inverse of Scalar Counterpart: a.a -1 1

21 Inverse of a 2x2 matrix How to invert a 2x2 matrix: 1. Swap the two numers on the lead diagonal a c d d 1 ad c 2. Change the sign of the off-diagonal elements c a 3. Multiply the matrix y the scalar d c a

22 Determinant of a 2x2 matrix Determinant of a matrix (2x2) case: Note: If det ( ) a c d a ad c c d -1 0 does not exist 1 Since does not exist 0 matrix with a non-zero determinant is said to e non-singular Example: Find the inverse of the following matrices. re they singular or non-singular? B

23 Solving Systems of Equations Matrices can e used to solve systems of equations From efore: To solve: Note: Cannot divide a matrix y a matrix instead multiply y inverse 4x + 3y 2x + x 11 y 5 Can e expressed in Matrix notation as : where Multply oth sides y : ( ) 1 1 x ( ) 1 1 x Ix x 3, x x, y 11 5 Rememer: multiplying a matrix y I has same effect as multiplying a scalar y 1

24 Solving Systems of Equations Example: Express the following systems in matrix form and solve for the unknown terms: a) ) 4x + 3y 2x + y 5 4P + P P 5P

25 pplication: National Income Determination Injections: Investment Injections: Government Investment Expenditure Y C+I* C FIRM S Y Expenditure: Consumption C of domestically produced goods Income: Payments for factors of production Ca+Y C H O U SEHOLD S Y W ithdrawals: : Savings Taxation Y C+S

26 pplication: National Income Determination Examples: The equilirium levels of income Y and consumption C for a simple two sector macroeconomic model satisfy the structural equations: YC+I* C a+y Where I* is planned investment. Write this system in matrix notation and solve for the equilirium level of income and consumption.

27 pplication: National Income Determination Examples: The equilirium levels of income Y and consumption for a simple two sector macroeconomic model satisfy the structural equations: YC+I* C Y where planned investment is I* 12 Write this system in matrix notation and solve for the equilirium level of income and consumption.

28 Cofactors of a matrix Each element of a matrix has a corresponding cofactor For a 3x3 matrix the cofactor ij is the determinant of the 2x2 matrix found y deleting the ith row and the jth column of and prefixing with a + or according to the following pattern

29 Cofactors of a matrix a a a a a a a a a Example: the cofactor 23 is found y 1. Deleting the 2 nd row and third column 2. Finding the determinant of the resulting matrix a a a a Prefixing with a + or - accordingly a a a a a a a a + a a ( ) a31 a32

30 Cofactors of a matrix Example Find all cofactors of the matrix:

31 Determinant of a 3x3 matrix To find the determinant of a 3x3 matrix we multiply the elements of any one row of the matrix y the corresponding cofactors and add them up It does not matter which row you use. det det det a11 a12 a13 a 21 a22 a 23 a31 a32 a33 ( ) ( ) a a a or + a or a ( ) a a a a

32 Determinant of a 3x3 matrix Example Find the determinant of the matrix:

33 Inverse of a 3x3 matrix The inverse of the matrix Is given y: a a a a a a a a a

34 Inverse of a 3x3 matrix Need to first compute the adjugate matrix (matrix of cofactors) Then the adjoint matrix (transpose of matrix of cofactors): Pre-multiply y: 1

35 Inverse of a 3x3 matrix Example Find the inverse of the matrix:

36 Using matrices to solve systems of equations Given the system of equations: a a a x + a x + a x + a y + a y + a y + a z 1 z 2 z 3 Write as: x a a a a a a a a a x x y z x 1

37 Using matrices to solve systems of equations Example Use matrices to solve the following system of equations: P P P P P P P P P

38 Cramer s Rule When solving any nxn system x the ith variale can e found using x i det det ( i ) ( ) Where i is the nxn n matrix found y replacing the ithi column of y the right hand-side vector Example: Given the system of equations x 5 x Find the value of x 2

39 Cramer s Rule 4 2 x 5 x x 2 ( ) ( ) det 2 det x ( ) 4( 6) 7( 5) 2( 4)

40 Cramer s Rule Example Use Cramer s Rule to solve the following system of equations for x x x x x x x x x x

41 pplication: National Income Determination Example: In a closed economy the consumption function is given y C Yd and planned investment is I* 35 the government spends 20 on goods and services ut charges levies a lump sum tax of 25 and a proportional tax of 20%. Write this system of equations in matrix notation and use Cramer s Rule to solve for the equilirium level of income?