Understanding Matrix Algebra: Solving Equations & Inverses

Lecturer: Dr. Monica Lambon-Quayefio, Dept. of Economics Contact Information: mplambon-quayefio@ug.edu.gh College of Education School of Continuing and Distance Education 2014/2015 2016/2017

Session Overview This session continues the study of matrix algebra, specifically focusing on how to find the solution to a system of equations using the inverse and determinants of matrices. In this session we consider the fundamentally important concept of inverse of a square matrix and its main properties in solving systems of linear equations. This session also discusses the application of the Cramer s rule in solving a system of n linear equations and n unknowns. Objectives: Understand and be able to determine the determinant of 2 X 2 matrix Understand and determine the matrix of co-factors of a 3X3 matrix Determine the inverse of square matrices Understand the properties of the inverse matrix Understand how the Cramer s Rule works Apply the Cramer s rule in solving systems of equations. Slide 2

Session Outline The key topics to be covered in the session are as follows: Matrix Inverses Determinants: 2x2 and 3x3 matrices Cramer s Rule and its Application Slide 3

Reading List Sydsaeter, K. and P. Hammond, Essential Mathematics for Economic Analysis, 2nd Edition, Prentice Hall, 2006- Chapter 16 Dowling, E. T., Introduction to Mathematical Economics , 3rdEdition, Shaum s Outline Series, McGraw-Hill Inc., 2001.- Chapter 11 Chiang, A. C., Fundamental Methods of Mathematical Economics , McGraw Hill Book Co., New York, 1984.- Chapter 5 Slide 4

Topic One MATRIX INVERSE Slide 5

Inverse: Multiplicative Identity The multiplicative identity for real numbers is 1. The property is written as: a x 1 = 1 x a = a In terms of matrices, we need a matrix that can be multiplied by a matrix (A) and give a product which is the same matrix (A) This matrix exists and it is called the identity matrix. It is named Iand it comes in different sizes It is a square matrix with all 1 s on the main diagonal and all other elements are 0 Slide 6

Examples of Identity The following gives the Identity matrices of a 2x2 matrix, a 3x 3 matrix and a 4x4 matrix respectively: 1 0 = 2I 0 1 1 0 0 = 0 1 0 3I 0 0 1 1 0 0 0 0 1 0 0 = I 0 0 1 0 0 0 0 1 Slide 7

Given the matrix A below multiply AI 2 5 = A 4 0 2 5 1 0 4 0 0 1

The identity Matrix for Multiplication Let A be a square matrix with n rows and n columns. Let Ibe a square matrix with the same dimensions with 1 s on the main diagonal and 0 s elsewhere Then AI = IA = A

The multiplicative Inverse For every non-zero real number a, there is a real number 1/a such that a(1/a)=1 In terms of matrices, the product of a square matrix and its inverse is I + + 3 1 1 1 ) 1 ( 3 ( 1 ) 2 ( 3 ) 1 ) 3 ( 1 1 0 = = + + 2 1 2 3 ) 1 ( 2 ( 1 ) 2 ( 2 ) 1 ) 3 ( 1 0 1

The inverse of a Matrix Let A be a square matrix with n rows and n columns In terms of matrices, the product of a square matrix and its inverse is I + + 3 1 1 1 ) 1 ( 3 ( 1 ) 2 ( 3 ) 1 ) 3 ( 1 1 0 = = + + 2 1 2 3 ) 1 ( 2 ( 1 ) 2 ( 2 ) 1 ) 3 ( 1 0 1

The inverse of a Matrix Let A be a square matrix with n rows and n columns. If there is an n x n matrix B such that AB = I and BA = I , then A and B are inverses of one another. The inverse of a matrix A is denoted by A-1. Inverses have the following properties: ( ) B AB = ( ) A A' = ( A A = 1 1 )' 1 -A - ( 1 1 - ) 1 1 -

Inverse of a Matrix To show that matrices are inverses of one another, show that the multiplication of the matrices is commutative and the results is the identity matrix. Example: Show that A and B are inverses of each other. 2 3 5 3 = = A and B 3 5 3 2

2 3 5 3 = AB 3 5 3 2 + + ) 5 ( 2 ( 3 ) 3 ( 2 ) 3 ) 2 ( 3 = + + ) 5 ( 3 ) 3 ( 5 ( 3 ) 3 ) 2 ( 5 1 0 = 0 1

5 3 2 3 = BA 3 2 3 5 + + ) 2 ( 5 ( ) 3 )( 3 ) 3 ( 5 ( ) 5 )( 3 = + + ) 2 ( 3 ) 3 ( 2 ) 3 ( 3 ) 5 ( 2 1 0 = 0 1

Finding the Inverse of a Matrix: Method 1 Use the equation AB = I 1 2 a b = = Let A and B 3 5 c d Write and Solve the equation 1 2 1 0 a b = 3 5 0 1 c d

1 2 1 0 a b = Multiply the two matrices to get the matrix below + 3 5 0 1 c d + 2 2 1 0 a c b d = Use matrix equality to equate corresponding elements + + 3 5 3 5 0 1 a c b d + a = + b = 2 1 2 0 a c b d Solve by substitution / elimination to obtain the elements of the inverse matrix. + = + = d 3 5 0 c 3 = 5 and 1 c d a b = = = 5 3 2 1 and

So the inverse of A is 5 2 3 1 We can check this my multiplying A x A-1 = 0 1 + + 1 2 5 2 ( 1 ) 5 ) 3 ( 2 ) 2 ( 1 ( 2 ) 1 ( 3 + + 3 5 3 1 ) 5 ) 3 ( 5 ) 2 ( 3 ( 5 ) 1 = 0 1

Properties of Inverses ( ) 1 = 1 1 -A - AB B ( )' ( ) 1 = 1 - A' A Slide 19

Topic Two DETERMINANTS Slide 20

Determinants Each matrix can be assigned a real number called the determinant of the matrix. It is denoted by the symbol d c b a a b = If A means the determinant of A d c

The determinant of a 2 x 2 matrix is found as follows: a b = ad cb c d Find the determinant of the matrix = G 7 8 6 7 7 8 = ) 8 ( 6 = = ) 7 ( 7 49 48 1 6 7

1 1 Find the determinant of the matrix = H 2 2 1 1 ) 2 ( 1 = ) 1 ( 2 = 0 2 2 If the determinant of a matrix is 0, the matrix does not have an inverse. The matrix is then said to be invertible

Method 2: Using determinants to find Inverse of Matrix a b = ( ) , 0 If A and det A then c d d b 1 = 1 A c a ( ) det A

is called the adjoint of the original matrix a c d b It is found by found by switching the entries on the main diagonal and changing the signs of the entries on the other diagonal.

Properties of Determinants Determinants have several mathematical properties which are useful in matrix manipulations. 1 |A|=|A'|. 2. If a row or column of A = 0, then |A|= 0. 3. If every value in a row or column is multiplied by k, then |A| = k|A|. 4. If two rows (or columns) are interchanged the sign, but not value, of |A| changes. 5. If two rows or columns are identical, |A| = 0. 6. If two rows or columns are linear combination of each other, |A| = 0 7. |A| remains unchanged if each element of a row or each element multiplied by a constant, is added to any other row. 8. |AB| = |A| |B| 9. Det of a diagonal matrix = product of the diagonal elements Slide 26

Find the multiplicative inverse of: 1 2 1 2 = ) 4 ( 1 = ) 2 ( 3 = First find the determinant 2 A 3 4 3 4 3 2 1 4 2 1 Next, find the adjoint and then use the formula for finding the inverse. = = 1 A 1 3 1 2 2 2

Practice Questions on Inverses. Find the inverses of the matrices below using determinants. 2 1 0 3 1 3 1 1 4 8 2 4

Determinant of a 3 x 3 Matrix One way to find the determinant of a 3 x 3 is the formula below: a b c e f d f d e = + d e f a b c h i g i g h g h i

Example Find the determinant of the matrix below using the formula 2 0 5 3 1 5 0 2 4 2 0 5 1 5 3 5 3 1 = + 3 1 5 2 0 5 2 4 0 4 0 2 0 2 4 ( ) = ) 5 ) 5 + ( 1 2 4 ) 2 ( 0 ( 3 4 ) 0 ( 5 ( 3 2 ) ) 1 ( 0 = + 2 14 ) 0 ( 12 ( 5 6 ) = 28 30 = 2

Minor of a matrix A minor |Mij|is the determinant of the sub-matrix formed by deleting the ?? row and ?? column of the matrix. Matrix A is given below as: a a A = a 11 12 13 a a a 21 22 23 a a a 31 32 33 Write out all the 9 minors associated with this matrix.

Cofactor of a matrix A cofactor |Cij| is a minor with a prescribed sign. The rule for the sign of a cofactor is |Cij|= ( 1)?+? |Mij| For example: |C11|= ( 1)1+1 |M11| No need to calculate each sign. Just note this: + + + + +

Inverse of a 3x3 matrix The inverse of a 3x3 matrix is determined by using the formula below: 1 B-1 = |B| (matrix of co-factors)T

FINDING THE INVERSE OF A 3X3 MATRIX 1 1 1 2 1 2 2 3 Calculate the inverse of B = 4 Find the co-factors: M11 = 2 2 |M11| = 2 C11 = 2 3 4

1 1 2 2 3 Calculate the inverse of B = 1 1 2 4 M12 = 1 2 |M12| = 0 C12 = 0 2 4

1 1 1 1 2 2 3 Calculate the inverse of B = 4 2 Find the co-factors: M13 = 1 2 |M13| = -1 C13 = -1 2 3

1 1 1 2 1 2 2 3 Calculate the inverse of B = 4 Find the co-factors: M21 = 1 1 |M21| = 1 C21 = -1 3 4

1 1 1 2 1 2 2 3 Calculate the inverse of B = 4 Find the co-factors: M22 = 1 1 |M22| = 2 C22 = 2 2 4

1 1 1 2 1 2 2 3 Calculate the inverse of B = 4 Find the co-factors: M23 = 1 1 |M23| = 1 C23 = -1 2 3

1 1 1 2 1 2 2 3 Calculate the inverse of B = 4 Find the co-factors: M31 = 1 1 |M31| = 0 C31 = 0 2 2

1 1 1 2 1 2 2 3 Calculate the inverse of B = 4 Find the co-factors: M32 = 1 1 |M32| = 1 C32 = -1 1 2

1 1 1 2 1 2 2 3 Calculate the inverse of B = 4 First find the co-factors: M33 = 1 1 C33 = 1 |M33| = 1 1 2

1 1 1 2 1 2 2 3 Calculate the inverse of B = 4 Next the determinant: use the top row: |B| = 1x |M11| -1x |M12| + 1x |M13| = 2 0 + (-1) = 1

Using the formula, 1 B-1 = (matrix of co-factors)T |B| 1 = (matrix of co-factors)T 1

Using the formula, 1 B-1 = (matrix of co-factors)T |B| T 1 2 0 2 -1 -1 1 1 = -1 0 1

Using the formula, 1 B-1 = (matrix of co-factors)T |B| 0 2 0 -1 -1 2 -1 -1 = 1

Topic Three CRAMER S RULE AND ITS APPLICATION Slide 47

Systems of Equations Matrices can be used to find the solutions of systems of equations. First, the system of equations must be put in the matrix form Consider a system of equations with two unknowns, ?1 and ?2. a?1+ b?2 = e c?1 + d?2 = f

This can be represented in matrix form as follows: Ax=B where the Matrix A is called the matrix of coefficients written as d c a b ?2 C= ? A= x= ?1 ? Cramer s rule is a method that uses determinants to solve a system of equations. This method was named after the Swiss mathematician Gabriel Cramer (1704-1752)

ax + by = e is (x,y) In general the solution to the system cx + dy = f b e f d x= b a c d b a where and = 0 c d e a If we let A be the coefficient matrix of the linear system, notice this is just det A. c f y= b a c d