1. Dr Huw Owens Room B44 Sackville Street Building Telephone Number 65891
TX-1037 Mathematical Techniques for Managers Lecture – Matrices in Economics
Dr Huw Owens
Room B44 Sackville Street Building
Telephone Number 65891

2. Matrices in Economics – Objectives 1
Appreciate that using matrices you can handle an array of numbers as a single entity
Understand the definitions of a matrix and a vector and be able to transpose them
Carry out matrix addition and subtraction
Multiply a matrix by a scalar or another matrix
Use matrices to handle a system of equations
Find the value of a determinant
Calculate the inverse of a matrix
Use an inverse to solve a system of equations
Perform matrix operations in Excel and use Solver to solve a system of equations
Dr Huw Owens - University of Manchester : 02/01/2019

3. Matrix: a rectangle of values arranged in rows and columns
Elements of a Matrix
Matrix: a rectangle of values arranged in rows and columns
Element: the value in a particular row and column of a matrix
Vector: a matrix with only one row, or one column
Dr Huw Owens - University of Manchester : 02/01/2019

4. Dimension and Transpose of a Matrix
The dimension of a matrix is rows ´ columns
The transpose of a matrix is found by interchanging rows and columns
Square matrix: a matrix with the same number of rows as columns
Order: the number of rows and columns of a square matrix
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5. If a square matrix equals its transpose it is symmetric
Symmetric Matrices
Principal diagonal: the elements in a square matrix that lie in the same row and column
Symmetric matrix: a square matrix where the elements above the principal diagonal are the mirror image of those below
If a square matrix equals its transpose it is symmetric
Identity matrix I: a square matrix with 1s in the principal diagonal and 0s elsewhere
Dr Huw Owens - University of Manchester : 02/01/2019

6. Matrix Dimensions
State the dimension of each of the following matrices or vectors, and write down the transpose of each. Are any of the matrices symmetric?
A=2x2, b=4x1, C=2x4, D=3x3
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7. Matrix Addition and Subtraction
Matrix addition and subtraction is applicable only if the matrices have the same dimension
Add or subtract corresponding elements in the matrices
Equal matrices: matrices with the same dimension and all corresponding elements equal
Null matrix: a matrix in which all the elements are 0
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8. Matrix Addition and Subtraction
Just as in ordinary arithmetic, matrix addition and subtraction lets us find sums of values and the differences between them.
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9. Matrix Addition and Subtraction
Notice that the elements of a matrix can be zero or negative.
The matrix operations of addition and subtraction apply to entire matrices. These must have the same dimension, otherwise there will be no corresponding elements in the other matrix.
If a matrix is subtracted from itself we obtain a matrix of the same dimension in which all the elements are zero. The resulting matrix is known as a null matrix and denoted 0. For example,
Dr Huw Owens - University of Manchester : 02/01/2019

10. Matrix Addition and Subtraction Questions
If possible, find A+B, A-B, A+C, B-C, C-C, C-0 where
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11. Matrix Addition and Subtraction Questions
A+C and B-C are not possible as they have different dimensions.
C-C is a square matrix of order 3, the same dimension as C.
For C-0 we write a null matrix of the same dimension as C.
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12. Scalar Multiplication
Scalar: a single value, not a matrix of values
If matrix A is multiplied by a scalar, k, each element in A is multiplied by k
kA = Ak
Commutative property: the order in which items that are connected by an arithmetic operator are written does not affect the result
For example,
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13. Scalar Multiplication
Find the scalar products kB and Bk for the matrix
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14. Matrix Multiplication
Conformable for multiplication: the matrices are of appropriate dimensions for their product to be formed
Pre-multiply A by B: B is written before A to form the product BA
Post-multiply A by B: B is written after A to form the product AB
Dr Huw Owens - University of Manchester : 02/01/2019

15. Checking if Matrices are Conformable
Write the dimension of each matrix under the proposed product
If the inner numbers of the dimensions are equal you can form the product, and the outer numbers give its dimension
can be formed because q = q
The product is of dimension n by k
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16. For each entry in the product matrix A B
Choose the appropriate row of A and column of B as shown in the table
multiply corresponding pairs of elements and sum the results
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17. If possible form AB and BA where
Example
If possible form AB and BA where
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18. No Commutative Property
The commutative (independence of order) property of multiplication does not hold for matrix multiplication
In general, A B ¹ B A, and one of these products may exist while the other does not
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19. Transpose of a Product
The transpose of a product is the product of the separately transposed matrices in the reverse order
(AB) ¢ = B¢ A¢
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20. Associative Property
Associative property: the result of arithmetic operations is not affected by groupings of the components
When the associative property holds different groupings give the same result:
A(BC) = (AB)C
This property holds for matrix multiplication providing the dimensions of the matrices are such as to permit the different products to be formed
Dr Huw Owens - University of Manchester : 02/01/2019

21. Distributive Property
Distributive property: one arithmetic operation can be distributed over another without changing the result
The distributive property allows us to multiply out brackets, distributing what is multiplying the brackets so that it multiplies the terms separately
We can state this as
A(B + C) = AB + AC
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22. Matrix Questions
(1) If k is a scalar, find 3a, kB and Ck for the matrices
For the above matrices (a, B and C), find
(i) B+C, a(B+C), aB, aC and hence show a(B+C)=aB+aC
(ii) BC, a(BC), (aB)C and hence show a(BC)=(aB)C
(2) If k is a scalar, find 5A, kB and 7x for the matrices
(3) For matrices A, B and x in Q2 above, find where possible, AB, BA, Bx and BTx
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23. (5) If find matrix expressions for uTu and uuT.
Matrix Questions
(4) For matrices A and x in Q2, find (Ax)T and xTAT and show they are equal.
(5) If find matrix expressions for uTu and uuT.
Find their numerical values if
For vectors x,y and 1 find the matrix expressions for xT1, yT1, xTy and yTx
Find the numerical value of the expressions if
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24. If X= show that XTX is a square symmetric matrix
Matrix Questions
Find XTy for
Show that AI=IA=A for
If X= show that XTX is a square symmetric matrix
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25. Matrix answers
(1)
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26. Matrix Answers Find 5A, kB and 7x
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27. Find, where possible, AB, BA, Bx and BTx
Matrix Answers
Find, where possible, AB, BA, Bx and BTx
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28. Equation System in Matrix Notation
n equations in k unknowns are written in matrix notation as
Ax = c
where
A is a n ´ k matrix of coefficients
x is a k ´ 1 vector of unknown variables and
c is a n ´ 1 vector of constants
Dr Huw Owens - University of Manchester : 02/01/2019

29. Equation Systems
For a solution, number of linearly independent equations = number of unknown variables
Linearly independent: the property that none of the equations are linear combinations of other equations
If one equation is a linear combination of other equations the system cannot be solved
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30. Linearly Independent Equations
With the equations in matrix form, the condition for linear independence is that there should be no linear relationship between the rows or between the columns of the coefficients matrix
If this is the case, the matrix is non-singular
A matrix that contains linearly dependent rows or columns is described as singular
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31. A determinant is a number, or scalar, associated with a square matrix
Determinants
A determinant is a number, or scalar, associated with a square matrix
It is found by combining the elements of the array according to certain rules
The determinant of matrix A is denoted
If = 0, matrix A is singular
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32. Determinant of a 2 ´ 2 Matrix
If A is the 2 ´ 2 matrix
Its determinant is
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33. + if (i + j) is an even number and – if (i + j) is an odd number
Cofactors
For a 3 ´ 3 matrix, the cofactor of aij is the determinant of the 2 ´ 2 matrix that remains when row i and column j are deleted, together with the appropriate sign
The sign is
+ if (i + j) is an even number and
– if (i + j) is an odd number
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34. Determinant of a 3 ´ 3 Matrix
Choose any row or column of A
Multiply each element in that row by its cofactor, and add these products
A determinant of zero indicates a singular matrix
But note that you get a zero value if you use the wrong cofactors to expand a matrix
Dr Huw Owens - University of Manchester : 02/01/2019

35. Solving Equations in Matrix Form
For a system of equations in the matrix form Ax = c
If we could find an expression for x it would comprise a solution to the set of equations
Division does not exist in matrix algebra so we cannot divide by A
Instead, we may be able to find a matrix called the inverse of A that will help us
The matrix is denoted A-1
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36. To have an inverse A-1, matrix A must be square and non-singular
Inverse of a Matrix
To have an inverse A-1, matrix A must be square and non-singular
The inverse has the property A-1A = AA-1 = I
Pre-multiply both sides of the equation by the inverse:
A-1Ax = A-1c so
Ix = A-1c and
x = A-1c
Dr Huw Owens - University of Manchester : 02/01/2019

37. interchange the elements on the principal diagonal
Inverse of a 2 ´ 2 Matrix
To invert the 2 ´ 2 matrix
interchange the elements on the principal diagonal
change the signs of the other two terms
multiply by the scalar
Dr Huw Owens - University of Manchester : 02/01/2019

38. Calculate the cofactors of the elements, place them in a matrix
Inverse of a 3 ´ 3 Matrix 1
To invert the
3 ´ 3 matrix A=
Calculate the cofactors of the elements, place them in a matrix
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39. transpose the cofactor matrix to find the adjoint of A Adj(A) =
Inverse of a 3 ´ 3 Matrix 2
transpose the cofactor matrix to find the adjoint of A
Adj(A) =
Calculate the determinant of matrix A,
Dr Huw Owens - University of Manchester : 02/01/2019

40. Inverse of a 3 ´ 3 Matrix 3
multiply Adj(A) by the reciprocal of the determinant to form the inverse
A-1 =
check that the inverse is correct
either post-multiplying or pre-multiplying it by the original matrix should give an identity matrix
Dr Huw Owens - University of Manchester : 02/01/2019

41. Matrix Operations in Excel
Set out each matrix in Excel just as you would write it on paper
Put each matrix element in a separate Excel cell
You can then perform matrix calculations, either entering formulae for yourself or using the inbuilt formulae that Excel provides
Dr Huw Owens - University of Manchester : 02/01/2019

42. Transposing a matrix
You can transpose a matrix in Excel by copying it and then using the ‘Paste Special’ facility
Before you start the procedure, consider whereabouts in the spreadsheet the transposed matrix is to be placed
Remember that unless the matrix is square the transposed matrix will have a different dimension to the original matrix
Dr Huw Owens - University of Manchester : 02/01/2019

43. Matrix multiplication
Check the matrices are conformable and find the dimension of the product matrix before using Excel
Select a rectangle of cells of the appropriate dimension for the product you are going to form
Click the function button, choose the function MMULT
Click on the boxes in turn and either type the name of the appropriate matrix or select its cells in the worksheet
Press Ctrl+Shift+Enter to enter the array formula
Dr Huw Owens - University of Manchester : 02/01/2019