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Matrices
Unit II

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Deemed to be University
Matrix Definition
● An array of numbers arranged in columns and rows within a
rectangular space enclosed by a square bracket is called the Matrix.
● The numbers are called the elements or entries of the matrix.
● Example 1:
4 2 5
2 1 3
1 6 4

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● Example 2:
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
● In the matrix, the subscripts represent the placement of the elements
in the matrix.
● In the matrix, the first subscript shows the row placement and the
second subscript refers the column placement of the elements under
reference.
● Thus 𝑎11 is the first row first column element and 𝑎32 is the third row
second column element and so on.

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Order of a Matrix
● The number of rows and columns determines the size of the given
matrix. It is often called the order of the matrix.
● For example, if the matrix contains three rows and four columns, then
it is called a three by four matrix and is denoted by 3 × 4.

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Vectors
● A vector is a special type of matrix in which there is only one row or
one column.
● If it contains one column, then it is called a column vector. On the
other hand, if it contains only one row, then it is called a row vector.
● Vector Addition: The sum of two vectors of the same type (i.e of
same order) is a third vector whose elements are the sums of the
corresponding elements of the given individual vectors.
● Scalar multiplication: if we want to multiply a vector by a scalar (like
any number), we simply multiply each and every element of the
vector by that scalar number)

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Elementary Operations in Matrices
● Addition / Subtraction
● Multiplication

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Matrix Addition
● Addition of two matrices: The sum of two matrices of same order is
a third matrix whose elements are the sums of the corresponding
elements of the given matrices.
Laws of Addition:
● Associative Law of Addition: When we add two matrices, it does not
matter in which order they are grouped together for addition, i.e.,
𝐴 + 𝐵 = 𝐵 + 𝐴.
● Commutative Law of Addition: When we add three or more matrices,
it does not matter in which order they are grouped together for
addition, i.e.,
𝐴 + (𝐵 + 𝐶) = (𝐴 + 𝐵) + 𝐶.

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Multiplication of a Matrix
● Scalar multiplication: When we want to multiply a matrix by a scalar,
then each and every element of the matrix gets multiplied by that
scalar.

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Multiplication of a matrix by another matrix
● Suppose A and B are the two matrices such that the number of
columns (say n) of A is equal to the number of rows (say m) of B.
● Now for to get the multiplied matrix denoted by 𝐴 × 𝐵,
○ We multiply the corresponding elements of the first row in A with the
corresponding elements of the first column elements of B and add them all
together to get first column first row element of the resulting matrix
● Thus, in general the ith element of 𝐴 × 𝐵 is obtained by multiplying
and adding corresponding elements of the ith row of A with
corresponding jth column elements of B.

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Commutative and Associative Axioms
● Commutative law of multiplication: The associative law of
multiplication is no longer true for the matrix multiplication. This is
to say that 𝐴 × 𝐵 = 𝐵 × 𝐴.
If A =
1 2 3
2 3 4
B =
1 2
2 1
2 4
, then, verify associative law of matrix
multiplication?
● Associative law of multiplication: The commutative law of
multiplication holds good provided the related products are defined.
𝐴 × 𝐵 × 𝐶 = (𝐴 × 𝐵) × 𝐶

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Matrix Multiplication is associative, but not
commutative
● Let A =
3 3
2 4
, B =
2 1
−1 1
and C =
3 1
1 −2
; find
𝐴 × 𝐵 × 𝐶 = (𝐴 × 𝐵) × 𝐶
● If A =
2 5
0 1
, B =
1 2 1
2 0 0
and C =
2 2
0 1
5 0
are the three
matrices. Calculate A x (B x C) and (A x B) x C, and show that they
are equal.

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Types of Matrices
● Square matrix: A matrix in which the number of rows is equal to the
number of columns is said to be a square matrix. Thus, an m × n
matrix is said to be a square matrix if m = n and is known as a square
matrix of order ‘n’.
● Rectangular matrix: A matrix is said to be a rectangular matrix if the
number of rows is not equal to the number of columns.
● Null matrix: A matrix with ‘zero’ elements everywhere is called the
Null matrix or zero matrix.
● Identity or Unit matrix: is a square matrix containing unity along the
main (principal) diagonal and zeros in other places, and it is
represented by I. The identity/unit matrix will act like unity in
ordinary matrix leaves the original matrix unaltered.

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● Diagonal matrix: A matrix with all elements zeros other than the
main or principal diagonal is called the diagonal matrix.
● Transpose of a matrix: If we interchange the rows and columns of an
m × n matrix A, we get n × m matrix. This newly obtained matrix is
called the transpose of the matrix A.
● Symmetric matrix: A symmetric matrix is a square matrix that is
equal to its transpose matrix. The transpose matrix of any given
matrix A can be given as 𝐴𝑇
. A symmetric matrix A, therefore,
satisfies the condition, 𝐴 = 𝐴𝑇.

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Determinant of a matrix
● If A stands for the square matrix, then the associated determinant is
defined and denoted by |A|.
If A =
2 4
1 3
then |A| =
2 4
1 3
● Value of a (2ⅹ2) determinant:
|B| =
𝑎11 𝑎12
𝑎21 𝑎22
= 𝑎11 × 𝑎22 − (𝑎21 × 𝑎12)
|B| =
1 2
3 4
= 1 × 4 − 3 × 2 = 4 − 6 = −2

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● Value of a (3ⅹ3) determinant: (Rule for sign attachment)
● The value of 3x3 determinant there is obtained by six possible ways.
It can be through either 1st or 2nd or 3rd row or through 1st or 2nd or 3rd
column.
● For this purpose, we must attach relevant signs for each and every
element in accordance with their respective placement in the given
determinant.

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● Rule for sign attachment: Add the subscripts that refer the placement
of the given number and attach a plus provided the result is even. On
the other hand, if the said sum is odd then attach a minus.
● For first row and first column element namely 𝑎11, we attach a positive sign
because 1 + 1 = 2.
● If it is second row third column element (i.e., 𝑎23), then we must attach a
negative sign because 2 + 3 = 5.
● In a similar way, we attach a plus or a minus depending upon their respective
positions in the given determinant.
|A| =
+ − +
− + −
+ − +
● fgf

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● Expansion of a 3x3 matrix: The 3x3 can be expanded in six possible
ways.
a) Through first row
b) Through second row
c) Through third row
d) Through first column
e) Through second column
f) Through third column

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Minor
● The minor of an element in a matrix is defined as the determinant
obtained by deleting the row and column in which that element lies.
For example, in the determinant
|A| =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
,
minor of the element 𝑎11 is denoted as 𝑀11,
𝑀11 =
𝑎22 𝑎23
𝑎32 𝑎33

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Find the value of a (3x3) determinant
● Find the value of |A|=
1 2 −2
4 1 2
2 −3 0
● a) through first row

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● The minor of an element in a matrix is defined as the determinant
obtained by deleting the row and column in which that element lies.
● A cofactor is a number that is obtained by eliminating the row and
column of a particular element which is in the form of a square or
rectangle. The cofactor is preceded by a negative or positive sign
based on the element’s position.

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● Singular and non-singular matrix: the square matrix A is said to be a
singular matrix provided its associated determinant is zero i.e. |A| = 0.
On the other hand, if |A|≠0 (i.e. non-zero), is not equal to zero, then
the associated matrix is called non-singular.
● Co-factors: A cofactor is a number that is obtained by eliminating the
row and column of a particular element which is in the form of a
square or rectangle. The cofactor is preceded by a negative or positive
sign based on the element’s position.
● Adjoint matrix: the adjoint matrix of A is obtained by replacing the
elements of A by its respective cofactors and then transposing it.

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Co-factors matrix
● If 𝐴 =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
, then the cofactors of the element 𝐴11, 𝐴12 and
𝐴13 are ,
● 𝐴11 =
𝑎22 𝑎23
𝑎32 𝑎33
, 𝐴12 = −
𝑎21 𝑎23
𝑎31 𝑎33
and 𝐴13 =
𝑎21 𝑎22
𝑎31 𝑎32

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Adjoint matrix
● If A =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
● Then, Cofactors matrix of A is
𝐴11 𝐴12 𝐴13
𝐴21 𝐴22 𝐴23
𝐴31 𝐴32 𝐴33
● Adjoint matrix is transpose of cofactors matrix of A,
adj A =
𝐴11 𝐴21 𝐴31
𝐴12 𝐴22 𝐴32
𝐴13 𝐴23 𝐴33

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Inverse of a square matrix
● In ordinary algebra, 1/𝑎 = 𝑎−1
is called the inverse of a. It is known
fact that 𝑎−1
× 𝑎 = 1. Multiplication of a number by its inverse thus
always equal to unity.
● In a similar way, if A is a square matrix, then there exist a matrix
called 𝐴−1
so that 𝐴−1
𝐴 = 𝐼. Here the matrix 𝐴−1
is called inverse of
A.
● Inverse of a matrix is calculated by using the formula,
𝐴−1
=
𝑎𝑑𝑗.𝐴
|𝐴|
or 𝑎𝑑𝑗. 𝐴
1
|𝐴|

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For calculating the inverse of a matrix,
● A matrix should be a square matrix
● A matrix should be non-singular matrix, for which determinant
should not be zero.

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Steps in calculating the matrix inverse
● Calculate the determinant for the given matrix and confirm that it is a
non-singular matrix.
● Calculate the cofactors matrix for the given matrix. The rule for the
sign of a cofactor is 𝐶𝑖𝑗 = (−1)𝑖+𝑗
|𝑀𝑖𝑗|
● Write adjoint of the given matrix.
● Divide the adjoint matrix by the determinant.
● Verify whether the product of the matrix with its inverse yields unit
matrix (𝐴−1
𝐴 = 𝐼).

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Example 1
● If 𝐴 =
2 3 4
4 3 1
1 2 4
, calculate 𝐴−1
.
● Determinant of 𝐴 = −5.
● Cofactor matrix of A =
10 −15 5
−4 4 −1
−9 14 −6
● adj of A =
10 −4 −9
−15 4 14
5 −1 −6
● 𝐴−1
=
−2 0.8 1.8
3 −0.8 −2.8
−1 0.2 1.2

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Linear Equations in Matrix Form
● In the simultaneous equation system given if the number of independent
equations is equal to number of unknowns then we can always represent the
system in matrix form.
● For example consider the following two equations and two unknowns system.
𝑎11𝑥1 + 𝑎12𝑥2 = 𝑐1
𝑎21𝑥1 + 𝑎22𝑥2 = 𝑐2
In matrix notation, this system can be written as
𝑎11 𝑎12
𝑎21 𝑎22
×
𝑥1
𝑥2
=
𝑐1
𝑐2
i.e.,
𝐴𝑋 = 𝐶

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Solving linear equations
● Linear equations can be solved using
○ Matrix inverse
○ Cramers’ rule

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Inverse Matrix Solution for linear equations
Let 𝑨𝑿 = 𝑪 stands for the system in matrix form.
𝑿 = 𝑨−𝟏𝑪

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DETERMINANT
● Each and every square matrix will have a determinant associated with
it. Such determinants on expansion will always give a scaler value.
● To distinguish the determinant from the corresponding associated
matrix, we use tow vertical lines on either side as show below in the
place of the square bracket.
● If A stands for the square matrix, then the associated determinant is
defined and denoted by |A|.
If A =
2 4
1 3
then |A| =
2 4
1 3
is called
the associated determinant of the matrix A

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● Minors of a given determinant:
● |A| =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
, then 𝑎11 =
𝑎22 𝑎23
𝑎32 𝑎33
and 𝑎12 =
𝑎21 𝑎23
𝑎31 𝑎33
are called the minors.
● Thus, in general the minor of the ith row and the jth column element
is obtained by deleting the ith row and jth column in the associated
determinant of the given matrix.

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● Cofactors: The minor of an element, with its correct sign assigned in
accordance with the alternative sign rule is called the cofactor of the
element under reference.
● If 𝐴𝑖𝑗 refers the cofactor of the ith row and the jth column element
then by the alternating sign rule, the sign to be attached is given by
the expression (−1)𝑖+𝑗.
● Example:
𝐴11 = (−1)1+1
𝑎11 = (−1)𝟐
𝑎11 = +𝑎11
𝐴32 = (−1)3+2
𝑎32 = (−1)𝟓
𝑎32 = −𝑎32

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Some Important Properties of Determinants
● Property I: The interchange of rows and columns does not affect the
value of a determinant. In other words, the determinant of a matrix A
has the same of its transpose 𝐴𝑇
, that is 𝐴 = 𝐴𝑇
.
Example: Let |A| =
2 4
3 5
= 2 × 5 − 3 × 4 = 10 − 12 = −2
Now, let us interchange the rows into columns,
|B| =
2 3
4 5
= 2 × 5 − 3 × 4 = 10 − 12 = −2

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● Property 2. The value of the determinant is zero if any two of its
columns (or rows) are identical.
Let |A| =
2 2
3 3
= 2 × 3 − 3 × 2 = 6 − 6 = 0
● Property 3. The interchange of any two rows (or two columns) will
alter the sign, but not the numerical value of the determinant.
Let |A| =
2 4
3 5
= 2 × 5 − 3 × 4 = 10 − 12 = −2
Now, let us interchange the first and second column,
|B| =
4 2
5 3
= 4 × 3 − 5 × 2 = 12 − 10 = +2

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● Property 4. The multiplication of all the elements in any one row or
column by some scalar k will change the value of the determinant k-
fold.
Let |B| =
4 2
5 3
= 4 × 3 − 5 × 2 = 12 − 10 = 2
Now, let us multiply the first row elements completely by the scalar 2,
then
|C| =
8 4
5 3
= 8 × 3 − 5 × 4 = 24 − 20 = 4 {2ⅹ2}

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● Property 5: The addition (subtraction) of a multiple of any row to
(from) another row will leave the value of the determinant unaltered.
The same holds true if we replace the word row by column in the
previous statement.
Let |A| =
𝑎 𝑏
𝑐 + 𝑘𝑎 𝑑 + 𝑘𝑏
= 𝑎 𝑑 + 𝑘𝑏 − 𝑏 𝑐 + 𝑘 = 𝑎𝑑 − 𝑏𝑐
● Property 6: If one row (or column) is a multiple of another row (or
column), the value of the determinant will be zero.
Let |A| =
2𝑎 2𝑏
𝑎 𝑏
= 2𝑎𝑏 − 2𝑎𝑏 = 0

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Determinants types
● Simple Determinant
● Jacobian: First order partial derivatives
● Hessian: Second order partial derivations

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Cramers’ rule
● In linear algebra, Cramer's rule is an explicit formula for the solution of a
system of linear equations with as many equations as unknowns, valid
whenever the system has a unique solution.
● It expresses the solution in terms of the determinants of the (square)
coefficient matrix and of matrices obtained from it by replacing one column by
the column vector of right-sides of the equations.

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● The system of simultaneous equations in two unknowns in its general form is
normally written as
𝑎11𝑥 + 𝑎12𝑦 = 𝑐1
𝑎21𝑥 + 𝑎22𝑦 = 𝑐2
● According to Cramer’s rule,
𝑥 =
𝑐1 𝑎12
𝑐2 𝑎22
𝑎11 𝑎12
𝑎21 𝑎22
𝑦 =
𝑎11 𝑐1
𝑎21 𝑐2
𝑎11 𝑎12
𝑎21 𝑎22

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INPUT – OUTPUT ANALYSIS
● Input-output is a novel technique invented by Wassily W. Leontief in 1951.
● Input-output analysis tells us that there are industrial interrelationships and
inter-dependencies in the economic system as a whole.
● The inputs of one industry are the outputs of another industry and vice versa,
so that ultimately their mutual relationships lead to equilibrium between
supply and demand in the economy as a whole.
● It is also known as “inter-industry analysis”.

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Main features of Input-output Analysis
● Firstly, the input-output analysis concentrates on an economy which is in
equilibrium. It is not applicable to partial equilibrium analysis.
● Secondly, it doesn’t concern itself with the demand analysis. It exclusively
deal with technical problems of production.
● Lastly, it is based on empirical investigation.

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Importance of input-output Analysis
● A producer can know from the input-output table, the varieties and quantities
of goods which he and the other firms buy and sell to each other. In this way
he can make the necessary adjustments and thus improve his position vis-à-
vis other producers.
● It is also possible to find out from input-output table the interrelations among
firms and industries about possible trends towards combinations.
● The repercussions of a prolonged strike, of a war and of a business cycle can
be easily perceived from the input-output table.
● It is used for national income accounting as it provides a more detailed
breakdown of the macro aggregates and money flows

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Technology Matrix
● Technology matrix shows direct productive interdependence of all the
sectors of the economy. Each element of the technology matrix is
called as technological coefficient.
● Elements of technology matrix are called technical coefficients and
are denoted as:
𝑎𝑖𝑗 =
𝑋𝑖𝑗
𝑋𝑗
● The technical coefficient 𝑎𝑖𝑗 is defined as a ratio of a product from
sector i that is required by sector j in order to produce one unit of its
product. It shows the maximum possible production of the sector j
regarding the quantity of available intermediate products of the sector
i.
● As the technical coefficient 𝑎𝑖𝑗 is higher, the direct interdependence
between the sectors i and j is stronger, and vice versa

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Hawkins-Simon Conditions
● It is definitely unrealistic situation If input-output matrix solution gives negative
outputs. Such a system is not viable system. Hawkins-Simon conditions
guard against such eventualities.
● The Hawkins–Simon conditions are necessary and sufficient as well for the
convergence of the solution to the same equilibrium output levels.
1. The determinant of the matrix must always be positive.
2. The diagonal elements i.e. (1 − 𝑎11), 1 − 𝑎22 , (1 − 𝑎33), …………….
(1 − 𝑎𝑛𝑛) of the matrix should be positive. In other words, elements 𝑎11,
𝑎22 and 𝑎33 should all be less than one. One unit of output of any sector should
use not more than 1 unit of its own output.
● If Hawkins-Simon conditions are not satisfied, no solution is possible.

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Example
● Let us assume that the economy consists of two sectors namely steel and
coal producing sectors.
● Total output produced by any industry/sector can be divided into self-
consumption, other industries demand requirements and final demand (from
household sector, government, external sectors, etc.).
Steel Coal
Final
Demand
Output
Steel 8 12 10 30
Coal 7 11 9 27

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Leontief Expression of Input-output table
𝑋1 = 𝑎11𝑋1 + 𝑎12𝑋1 + 𝐹1
𝑋2 = 𝑎21𝑋1 + 𝑎22𝑋2 + 𝐹2
𝑋1
𝑋2
=
𝑎11 𝑎12
𝑎21 𝑎22
𝑋1
𝑋2
+
𝐹1
𝐹2
𝑋 = 𝐴𝑋 + 𝐹
𝑋 − 𝐴𝑋 = 𝐹
𝑋 1 − A = F
𝑋 = (1 − 𝐴)−1𝐹
Steel Coal
Final
Demand
Output
Steel 8 12 10 30
Coal 7 11 9 27

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Exercise 1
● Given input-output table as follows;
● Find the gross output requirements for each industry for the final demands 18
and 44 units respectively.
Input
Output
Industry I II Final
Demand
I 16 20 4
II 8 40 32

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Step 1 Complete the input-output table:
Input
Output
Industry I II
Final
Demand
Output
I 16 20 4 40
II 8 40 32 80

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Calculate technological coefficient matrix
● 𝐴 =
16/40 20/80
8/40 40/80
=
0.4 0.25
0.2 0.5
𝑋 = (𝐼 − 𝐴)−1𝐹

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Calculate (I-A) and its inverse
●
1 0
0 1
−
0.4 0.25
0.2 0.5
=
0.6 −0.25
−0.2 0.5
● (I-A) =
𝟎. 𝟔 −0.25
−0.2 𝟎. 𝟓

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Determinant of (I-A)
|I-A| = (0.6X0.5) – (-0.2X-0.25)
= 0.3 – 0.05 = 0.25
Since determinant value is positive and diagonal values are positive and less
than one.
Check the Hawkins-Simon conditions are satisfied or not.

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Inverse of I-A
(𝐼 − 𝐴)−1
=
0.6 −0.25
−0.2 0.5
−1
=
1
0.25
0.5 0.25
0.2 0.6
=
2 1
0.8 2.4

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𝑋 = (𝐼 − 𝐴)−1𝐹
𝑋 =
2 1
0.8 2.4
18
44
=
2 × 18 + (1 × 44)
0.8 × 18 + (2.4 × 44)
=
80
120
● If final demand for industries I and II increases to 18 and 44, each
industry should produce 80 and 120 respectively.

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Exercise 2:
● Given the following transaction matrix, find the gross output to meet the final
demand of 200 unit of agriculture and 800 units of industry:
● Refer Exercise 1 of Input-output analysis and solve this problem.
○ Complete the input output table
○ Calculate technology matrix and subtract from identity matrix
○ Verify Hawkins-Simon conditions (determinant of I-A & diagonal values)
○ If H-S are satisfied, calculate inverse of I-A and multiply with new final demand vector
matrix.
Producing
Sectors
Purchasing Sections Final
Demand
Agriculture Industry
Agriculture 300 600 100
Industry 400 1200 400

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● In an economy of three industries A, B & C, the data given below (in
millions) are available.
● Determine the output of each industry if the final demand changes to
60 for A, 40 for B and 60 for C.
Producers
Users
Final Demand
Out
put
A B C
A 80 100 100 40 320
B 80 200 60 60 400
C 80 100 100 20 300

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