Brief review on matrix Algebra for mathematical economics

INFO-LINK UNIVERSITY COLLEGE
Course Name: Mathematical Economics
By:Feleke Ph.(MSc)

Chapter One: A Brief Review Of Matrix Algebra
• Definition of Matrix
• A matrix is defined as a two-dimensional array of numbers (or
algebraic symbols) set out in rows and columns.
• The elements of a matrix can be numbers, variables, or symbols.
The elements are indexed by the row and then the column in which
they occur.
• For example, represents the item of matrix A located on the
second row, second column. The size of a matrix – called the order
– is denoted by the number of rows and then the number of
columns.

Cont…
 For instance a 2x3 matrix is a matrix that has two rows and three columns.
 A vector is a matrix that has only a single row or a matrix that has only a single
column.
 A column-vector has only a single column, such as:

Cont…
• If we don‟t specify row or column, we assume that
a vector is a column vector.
• A scalar is a matrix (or row-vector or column-
vector) that has only a single element.
• Scalars are nondimensional numbers, variables, or
symbols. For example, [1] or [a] or [฀ ] are all

Special Matrices
1. Identity Matrix: The square matrix of order n
with all diagonal elements equal to one, and all
offdiagonal elements equal to zero is called the
identity matrix of order n and is denoted as . The
identity matrix is symmetric and diagonal.

Cont..
• The identity matrix has a property similar to
the scalar number 1.
• Any matrix multiplied by the identity matrix
is equal to itself. This is shown below.
• [I] is the identity matrix and [M] is any other
matrix.
• [I]x[M]=[M]

Cont…
• 2. Symmetric Matrix: When A’= A, the matrix is called
symmetric. That is, a symmetric matrix is a square matrix, in
that it has the same number of rows as it has columns, and the
off-diagonal elements are symmetric

Cont…
• 3. Diagonal matrix: A square matrix with all
o¤-diagonal elements equal to zero is called a
diagonal matrix. A diagonal matrix is
symmetric.
Example :

Cont…
• 4. Triangular matrix: A square matrix with all elements below
the diagonal equal to zero is called an upper triangular matrix.
Similarly a matrix with all elements above the diagonal equal
to zero is called a lower triangular matrix.
 In the lower triangular matrix all the entries above the main diagonal are zero,
whereas
 in the upper triangular matrix all the entries below the main diagonal are zero.
 (b) Any diagonal matrix is both upper and lower triangular

Cont…
• 5. Null or zero matrix: The null or zero matrix is a
matrix with each element being zero. It is denoted
as 0.

Cont…
• 6. Idempotent matrix: Let A be a k × k matrix, A is
idempotent if AA = A
• 7. Orthogonal matrix: A square matrix A is orthogonal if A'A
= AA' = I

Matrix Operations
• Vector Arithmetic
• For vector arithmetic, we have to make sure that the vectors are
conformable for that particular operation.
• 1. Addition (& Subtraction): To be conformable, the vectors must
both be the “same size” (that is, they must both be row-vectors with
the same number of columns or both be column-vectors with the
same number of rows).

Cont…
• 2. Multiplication: To be conformable, we can
either multiply a row-vector times a column-
vector or a column-vector times a row-vector, but
we cannot multiply two row vectors or two
column vectors.
• Unlike scalar multiplication, the order is
important: RC ≠ CR

Cont…
• i. Inner Product: row-vector * column-vector = scalar. In
addition, to be conformable for innerproduct multiplication,
the number of columns in the row-vector must equal the
number of rows in the column-vector.

Cont…
• ii.Outer Product: column-vector * row-vector =
matrix.
• The vectors may be of any size, but their sizes
will determine the size of the matrix yielded by
their outer product.
•

Cont…
• We can generalize the formula by indicating the formula for
each element of the product. If we call the product matrix M,
mij refers to the element in the ith row and j ]th column of M.
• NB: There is no convenient “division” operator for vectors.

Matrix Arithmetic
• For matrix arithmetic, we must also make sure the matrices are
conformable as well.
• 1. Addition (& subtraction): Two matrices can be added if
they are conformable for addition, i.e., they have the same
number of rows and columns. Addition is performed by adding
(or subtracted as the case may be) together the corresponding
elements of each matrix and placing the results in the
corresponding elements of a new matrix of the same
dimensions.
• For example

Cont…
• 2. Multiplication: To be conformable for multiplication, the
number of columns of the first matrix must equal the number of
rows of the second matrix.
• The resulting product will have as many rows as the first matrix
and as many columns as the second matrix. As with vector
multiplication, the order of multiplication is important.
• The most elegant way to express the process of matrix
multiplication is to say that (i,j) element is given by the inner
product of the ith row of the first matrix and the jth column of
the second matrix.

Cont…
• Laws of Matrix Algebra: Associative Laws: (A+B)+C =
A+(B+C)
• (AB)C = A(BC)
• Commutative Law for Addition: A + B = B + A. Note
however that this law does not hold for multiplication, AB ≠
BA.

Determinant of a Matrix
• The determinant is a function that takes a square matrix as an
input and produces a scalar as an output.
• For a 2nd order matrix (i.e. order 2×2) the determinant is a
number calculated by multiplying the elements in opposite
corners and subtracting.
• The usual notation for a determinant is a set of vertical parallel
lines either side of the array of elements, instead of the squared
brackets used for a matrix. The determinant of the general 2 ×
2 matrix A, written as |A|, will therefore be as follows:
• Example:
• Find the determinant of the matrix

Cont…
• Solution Using the formula defined above, the
determinant of matrix A will be

Cont…
• The determinant of a 3rd order matrix
• The determinant of a 3x3 matrix is found the same way, except
that the determinant of the reduced matrix after eliminating the
row & column is the determinant of a 2x2 matrix rather than the
degenerate case previously when it was the determinant of a
scalar.
• This is a Laplace expansion of the matrix A along its first row.
• The determinant of a matrix can be evaluated with a Laplace
expansion along any one of its rows or any one of its columns.

Cont…
• The determinants of larger matrices are found the same way,
except that it takes more steps to find the determinant of the
reduced matrix.
• Since we define determinants are linear combinations of the
determinants of smaller sub-matrices, we call the algorithm a
“recursive” algorithm.
• The Laplace expansion method, discussed here in the context of a
3x3 matrix, can be expressed in another way that enables us to
generalize this method to matrices of higher dimensions.

Cont…
• Although the determinants of the 3rd order matrices above were
found by expanding along the first row, they could also have been
found by expanding along any other row or column.
• The same principle of multiplying each element along the expansion
row (or down the expansion column) by the determinant of the
matrix remaining when the corresponding row and column are
deleted from the original matrix A is employed.
• This can help make the calculations easier if it is possible to expand
along a row or column with one or more elements equal to zero.

Cont…
• However, there are rules regarding the sign of
each term, which must be followed.
• For a 3rd order determinant it is sufficient to
remember that the first term will be positive if
you expand along the 1st or 3 rd row or column
and the first term will be negative if you expand
along the 2nd row or column.
• The signs of the subsequent terms in the
expansion will then alternate.

Minors, Cofactors, Adjoint, Transpose
and Inverse of a Matrix
• Minor: A minor is the determinant of a
square submatrix of the matrix A. Let A
be an n x n matrix.
• Let Mij be the (n-1)x(n-1) matrix
obtained by deleting the i th row and
the j th column of A. The determinant
of that matrix, denoted |Mij|, is called
the minor of aij.

Cont…
• For example, minor of 3x3 matrix can be
determined as follows:
e h
h i
The minor of a(M11) is
determined as:
 The minor of the first element of the first row of the above matrix
A has been obtained after ignoring the first row and first column
of the above matrix and forming a new matrix.

Cont…
• Cofactor: The cofactor (denoted Cij) of the
element aij of any square matrix A is times
the minor of A that is obtained by including all
but the row and the column, or alternatively
the minor that is obtained by deleting the
row and the column.

Cont…
• Let A be an n x n matrix. The cofactor Cij is a
minor multiplied by either 1, if (i + j) is an even
integer, or – 1 if (i + j) is an odd integer.
• That is,
• The Laplace expansion can be expressed in a
compact way using cofactors. For example, the
evaluation of the determinant of A through a
Laplace expansion along its first row can be
written as

Cont…
Example: Compute the matrix of cofactors for the
given matrix.
(a) The minors of A are:
M11(A) = 2, M21(A) = 1, M12(A) = −1, M22(A) = 1,
And its cofactors are:

Cont…
• Thus, the matrix of cofactors for A is
(b) The minors of B are

Cont…
 The confactors of B are

Cont…
• Transpose: A matrix transpose is created by
swapping the rows and columns of a matrix. The
transpose of a k x n matrix A is the n x k matrix
obtained by interchanging the rows and columns
of A (it is typically denoted as or A’). To
transpose a matrix (or vector), each row is made
into a column, which means that each column
will become a row in the transposed matrix.

Cont…
• To transpose a matrix (or vector), each row is made into a
column, which means that each column will become a row in
the transposed matrix.
 If A = A’, then we say that A is symmetric.
 Thus, only square matrices may be symmetric since the sizes would
not work out for rectangular matrices.

Cont…
• Inverses: The matrix analogue of division is multiplying by an
inverse (which is another way of thinking about division in
scalar arithmetic). Finding the inverse of a matrix is
(relatively) easy for square matrices and (relatively) complex
for rectangular matrices.
• For scalars, the inverse of a number times that number is one.
A matrix inverse times that matrix should be equal to “one” –
but we need to find a matrix version of “one.”
• The matrix equivalent of one is called the “Identity Matrix.”
The identity matrix has one‟s on the main diagonal (elements
with row and column indices that are equal) and zeroes
elsewhere.

Cont…
• Let‟s check to make sure that
The general rule for inverse matrix is
Where, |A| is Determinant(det) of A

Cont…
 Compute the inverse of the given matrix A

Cont…
• Adjoint of a Matrix
• Let A be an n x n matrix. Define the n x n matrix in which the (i, j) th
element is the cofactor Cij of A as the matrix of cofactors.
• The adjoint matrix is an n x n matrix that is the transpose of the
matrix of cofactors.

Cont…
• Example: Compute the adjoints of the given
matrices

Solutions to Linear Simultaneous Equations
• Different economic problems can be interlinked in the real world and
may require a system solution as they can‟t be solved separately.
• Most of such inter-relationships are usually given in a linear form
which makes their solutions relatively easy.
• Among others, inversion method and the Cramer‟s rule are two of the
widely used methods to solve a system of simultaneous linear
equations.

Cont…
• Inversion Method
• The Inverse method is one of the important methods to solve a
linear system with n equations in n unknown.
• Use matrix algebra to solve for the unknown variables x1, x2 and
x3 given that:
 Solution
 This set of simultaneous equations can be set up in matrix format as
Ax = b where:

Cont…
 To derive the vector of unknowns x using the matrix formulation
x = we first have to derive the matrix inverse
 The first step is to derive the cofactor matrix, which will be:
 The adjoint matrix will be the transpose of the cofactor matrix and so :
 The determinant of A, expanding along the second row, will be:

Cont…
 The matrix inverse will therefore be:

Cont…
• Cramer’s Rule
• Cramer‟s rule is another method of using matrices for solving sets of
simultaneous equations, but it finds the values of unknown variables
one at a time.
• This means that it can be quicker and easier to use than the matrix
inversion method if you only wish to find the value of one unknown
variable.
• We already know that a set of n simultaneous equations involving n
unknown variables x1, x2, . . . , xn and n constants values can be
specified in matrix format as Ax = b where A is an n × n matrix of
parameters, x is an n × 1 vector of unknown variables and b is an n ×
1 vector of constant values.

Cont…
 The matrix of which we compute the determinant in the numerator of
the first expression is the matrix A, where the first column has been
replaced by the b vector.
 The matrix of which we compute the determinant in the numerator of
the second expression is the matrix A where the second column has
been replaced by the b vector. This procedure for solving systems of
equations is called Cramer’s rule.
 Cramer’s rule says that the value of any one of the unknown variables
xi can be found by substituting the vector of constant values b for the
ith column of matrix A and then dividing the determinant of this new
matrix by the determinant of the original A matrix.

• Thus, if the term Ai is used to denote matrix A with
column i replaced by the vector b then Cramer‟s
rule gives:
Example
Find x1 and x2 using Cramer‟s rule from the following set of
simultaneous equations
5x1 + 0.4x2 = 12
3x1 + 3x2 = 21

Cont…
• Solution
• These simultaneous equations can be represented in matrix format
as:
 Using Cramer‟s rule to find x1 by substituting the vector b of constants for column 1 in
matrix A gives :
 In a similar fashion, by substituting vector b for column 2 in
matrix A we get:

Brief review on matrix Algebra for mathematical economics

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