ECON 213 Elements of Mathematics for Economists Session 5: Introduction to Matrix Algebra- Part Two Lecturer: Dr. Monica Lambon-Quayefio, Dept. of Economics Contact Information: mplambon-quayefio@ug.edu.gh

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.

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

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

Topic One matrix inverse

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 I and it comes in different sizes It is a square matrix with all 1’s on the main diagonal and all other elements are 0

Examples of Identity The following gives the Identity matrices of a 2x2 matrix, a 3x 3 matrix and a 4x4 matrix respectively:

Given the matrix A below multiply AI

The identity Matrix for Multiplication Let A be a square matrix with n rows and n columns. Let I be 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

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

The inverse of a Matrix Inverses have the following properties: 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:

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.

Finding the Inverse of a Matrix: Method 1 Use the equation AB = I Write and Solve the equation

Multiply the two matrices to get the matrix below Use matrix equality to equate corresponding elements Solve by substitution / elimination to obtain the elements of the inverse matrix.

So the inverse of A is We can check this my multiplying A x A-1

Properties of Inverses

Topic Two determinants

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

The determinant of a 2 x 2 matrix is found as follows: Find the determinant of the matrix

Find the determinant of the matrix 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

is called the adjoint of the original matrix 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

Find the multiplicative inverse of: First find the determinant Next, find the adjoint and then use the formula for finding the inverse.

Practice Questions on Inverses. Find the inverses of the matrices below using determinants.

Determinant of a 3 x 3 Matrix One way to find the determinant of a 3 x 3 is the formula below:

Example Find the determinant of the matrix below using the formula

Minor of a matrix A minor |Mij| is the determinant of the sub-matrix formed by deleting the 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column of the matrix. Matrix A is given below as: 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) 𝑖+𝑗 |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: B-1 = 1 |B| (matrix of co-factors)T

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

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

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

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

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

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

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

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

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

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

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

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

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

cramer’s rule and its application Topic Three cramer’s rule and its application

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, 𝑥 1 and 𝑥 2 . a 𝑥 1 + b 𝑥 2 = e c 𝑥 1 + d 𝑥 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 A= x= 𝑥 1 𝑥 2 C= 𝑒 𝑓 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 In general the solution to the system is (x,y) cx + dy = f e b f d x= a b c d where a b c d = 0 and a e c f If we let A be the coefficient matrix of the linear system, notice this is just det A. y= a b c d

Example Solve the following system of equations using Cramer’s rule: 8x+5y=2 2x-4y=-10 Solution: The coefficient matrix is and its determinant is so and

The solution to the system of equations is (-1,2)

Cramer’s Rule for 3x3 Matrix Let A be the coefficient matrix of this linear system: If det A is not 0, then the system has exactly one solution. The solution is:

Lets solve this system equations by Cramer’s rule 2x – 3y + z = 5 x + 2y + z = -1 x – 3y + 2z = 1 Need to find the determinants of

Find the determinant of the Coefficient Matrix We will use this for the denominators in the all the fractions.

Solving for x Replace the x column with the answers. So

Solving for y Replace the y column with the answers. So

Solving for z Replace the z column with the answers. So

Session Problem Sets Use Cramer’s rule to solve the unknowns in the system of linear equations given below A) 2 𝑥 1 + 4 𝑥 2 - 3 𝑥 3 =12 3 𝑥 1 - 5 𝑥 2 +2 𝑥 3 =13 - 𝑥 1 +3 𝑥 2 + 2 𝑥 3 = 17 B) 11 𝑝 1 - 𝑝 2 - 𝑝 3 =31 - 𝑝 1 +6 𝑝 2 -2 𝑝 3 = 26 - 𝑝 1 -2 𝑝 2 + 7 𝑝 3 = 24 C)

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