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ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation PDF Download

MATRICES
LEARNING OBJECTIVES :

At the end of this chapter, you will be able to:

  • Definition of Matrix
  • Types of Matrix
  • Addition and Subtraction of Matrix
  • Multiplication of Matrices
  • Transpose of Matrix
  • Inverse of a Square Matrix
  • Adjoint of a Square Matrix
  • System of Linear Equations
  • Cramer’s Rule (Determinant Method)

INTRODUCTION

Matrices applications are used in Business, Finance and Economics. Matrices applications are helpful to solve the linear equations with the help of this cost estimation, sales projection, etc., can be predicted. In this chapter, we shall find it interesting fundamental applications of matrices.

Matrix:

Ram, Sita and Laxman are three friends. Ram has 5 books, 3 pencils and 2 pens. Sita has 10 books, 8 pencils and 5 pens. Laxman has 15 books, 10 pencils and 2 pens. The above information about three friends can be represented in the following form:


BooksPencilPen
Ram532
Sita1085
Laxman5102

We can express the above information in the following form:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
An arrangement or display of information in the above form is called a matrix.

Matrix (Definition)

A rectangular array of numbers (real/complex) denoted by:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

A is rectangular matrix with m rows and n columns. The numbers aij, i = 1,2 ……..m; j = 1,2,…..n of this array are called its elements aij, is associated. We shall denote a matrix either using by using brackets

[ ]; or ( ).

Notes:

  1. It is to be noted that a matrix is just an arrangement of elements without any value in rows and columns.
  2. The plural matrix is matrices.
  3. It is to be noted that a matrix is just an arrangement of elements without any value in rows and columns.

Order of a Matrix: A matrix A with m rows and n columns is called a matrix of order (m, n) or m × n (read as m by n).
Consider the matrix ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

It is a matrix of order 3 × 3. Here the 9 occurs in the third row and second column. The elements 5 occurs in the second row and third column. Thus in notations we may write: a32 = 9 and a23 = 5.

TYPES OF MATRICES

Row Matrix:

A matrix which has only one row is called a row matrix or row vector.

The matrices of the type [a1, a2, a3 ……..,an]; [1, 2, 5] are examples of row matrices.

Column Matrix:

A matrix which has only one column is called a column matrix or a column vector.
The matrices are of the types ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation are examples of column.

Zero Matrix or Null Matrix:

If every element of a m × n matrix is zero, the matrix is called zero matrix or null matrix of order (m, n) and it is denoted by : Omn. Thus [0, 0]; ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation are all zero matrices, but all of matrices,

Square Matrix and Rectangular Matrix: If the number of rows and columns in a matrix are same, such a matrix is called a square matrix; otherwise it is called a rectangular matrix.ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation are examples of square matrix are zero except on the leading diagonal, then it is called diagonal matrix.
Thus, if ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Is an n × n matrix, then the diagonal matrix obtained from it will be following type:
Diagonal Matrix = ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Scalar Matrix:

A diagonal matrix whose leading diagonal elements are all equal is called a scalar matrix, for example,
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Unit Matrix:

A scalar matrix whose diagonal elements are equal to unity is called unit matrix and it is denoted by In×n, if it is order of order n .For example,
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Upper triangle matrix:

A matrix is known as upper triangular matrix if all the elements below the leading diagonal are zero.

For example.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Lower Triangular Matrix:

A matrix is known as lower triangular matrix if all the elements above the leading diagonal are zero.

For example.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Sub Matrix:

The matrix obtained by deleting one or more rows or columns or both of a matrix is called its sub matrix. For example
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
The sub matrix is obtained by deleting second row and the second column from matrix A.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Equal Matrices:

Two matrices A=[aij] and B=[bij] are said to be equal if they satisfy the following two conditions.

(i) The order of both the matrices is same;

(ii) Corresponding elements in both the matrices are equal i.e.,

aij = bij (i = 1,2,…..m and j=1,2…….n)

ALGEBRA OF MATRICES

Addition and Subtraction of matrices: Let A and B be two matrices of the same order. Then the addition of A and B, denoted by A+B, is the matrix obtained by adding corresponding entries of A and similarly to subtract two matrices we just subtract their corresponding elements.

Thus, if A = (aij)m×n and B = (bij)m×n, then

A+B= (aij + bij)m×n

Remark: We can add two matrices of the same order. If they are of the same order, we say they are comfortable for addition. Also, the order of the matrices is the same as that of the two original matrices.

Property: If A, B, C are matrices of same order, then

(i) A + B = B + A (Commutative Law)

(ii) (A + B) + C = A + (B + C) (Associative Law)

(iii) K (A + B) = K.A + K.B, where m is constant
Example 1:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Negative of a Matrix: If A is any matrix, the negative of A is denoted by
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Scalar Multiplication

The multiplication of a matrix by scalar k implies the multiplication of every element.

Example 2:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Solution: Then k ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Multiplication of two matrices.

The product A B of two matrices A and B defined only if the number of columns in Matrix A is equal to the number of rows in Matrix B.

Properties of matrix Multiplication:

(i) Matrix multiplication is not commutative in general, i.e. AB ≠ BA.

(ii) Matrix multiplication is associative (AB) C = A(BC), where both sides are defined.

(iii) Multiplication distributes over addition of Matrices i.e.,

(a) A (B + C) = AB + AC

(b) (A + B) C = AC + BC

(iv) If A, B and C are three matrices such that AB = AC , then the general B ≠ C.

(v) If A is m×n matrix and O is an n × p null matrix, then AO = O, A= O

(vi) If A is a square matrix and I is a unit matrix of the same order, then AI = IA = O

(vii) Product of the two no-zero matrices is non zero matrix.
Example: 3 if ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundationfind AB
Solution: ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Example 4: The annual sale volume of three products X , Y , Z whose sale prices per unit are ₹ 3.50, ₹2.75 , ₹ 1.50 respectively. In two different market I and II are shown below.

MarketProduct
XYZ
I6000900013000
II12000600017000

Find the total revenue in each market with the help of matrices.

Solution: Let P denotes the column matrix of prices and S denote the rectangular matrix of sale of volumes of three different commodities at three different markets. Then,
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Total revenue in markets I and IIICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Transpose of Matrix: The matrix is obtained by interchanging rows and columns of a matrix A is called its transpose. Transpose of a matrix by AT or A’.

Symbolically, if A=[aij] and A’=[bij]

Then aij=bij
Example:
Let A= ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Properties of transpose of a Matrix:

(1) A matrix is transpose of its matrix i.e. A = (A’)’.

(2) The transpose of the sum of the two matrices is the sum of their transpose matrices, i.e.

(A + B)’= A’ + B’

(3) Transpose of a multiplication of a matrix and constant number is equal to the multiplication of the constant number by the transpose of matrix, i.e. (KA)’ = K.A’

(4) The transpose of the two matrices are equal to the product of their transpose in reverse order, i.e., (AB)’ = B’. A’

Symmetric Matrix: In any matrix if A is called symmetric then A’ = A.

Example 5: Let A = ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Here A’= A, A is called symmetric matrix.

Solution: Skew Symmetric matrix: Any matrix A is called skew symmetric. If A’=-A,

for a skew symmetric matric A = [aij], aij = – aij
Example 6: Find A2 , if ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Solution: A2 = A.A = ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Example 7: (a) Show that the matrix ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation satisfi es the equation:
A3 + 2A2 - A- 20 I = 0 .
Solution: 
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Now A3+2A2- A - 20I
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
(b) If A = ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation then show that A2 -(a+d)A = (bc - ad)I
Solution: We may write,
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
L.H.S. of the given equation = A2−(a+d)A
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

= (bc−ad) I = R.H.S.

Hence, A2 – (a + d) A = (bc–ad) I.

Example 8: If  ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation and (A+B)2 =A2 + B2, find the value of a and b
Solution:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation 
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Now (A + B)2 = A2 + B2
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Example 9: A company employs 60 labourers from either of party A and B, comprising of persons in different age groups as under:

Category
I (20-25 years)
II (26-30 years)
III (31-40 years)
Party A
252515
Party B
203010


Category
Rates
I1200
II1000
III600

Rate of Labour applicable to categories I, II and III are ₹1,200, ₹ 1,000 and ₹ 600 respectively. Using matrices, find which party is economically preferable over the other.
Solution:

Category
I (20-25 years)
II (26-30 years)
III (31-40 years)
Party A
252515
Party B
203010

Rates :

Category
Rates
I1200
II1000
III600

Labour charges payble ton each party can be given by
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Therefore, Party B is more economical as compared to Party A.

Example 10: In a certain city there are 5 colleges and 20 schools. Each school has 3 peons, 1 clerk and 1 head-clerk, where a college has 5 peons, 3 clerks, 1 head. clerk and additional staff of a caretaker. The monthly salary of a employee is as follows.

Peon ₹ 1100
Clerk ₹ 1700
Head-clerk ₹ 3000
Caretaker ₹ 2500

Using matrix method, find the total monthly bill of each college.
Solution: Let us put the information in tabular form:


PeonClerkHead-clerkCaretaker
School3110
College5311

ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Therfore, The monthly bill for a school and a college is given as:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Therefore, the monthly bill for a school is ₹8,000 and for a college is ₹16,100


Example 11: There are two families A and B .There are 4 men, 6 women and 2 Children in a Family A and 2 men, 2 women, and 4 children in Family B .The recommended requirement of calories in Man: 2400, Woman : 1900, Child : 1800 and for proteins in Man: 55 gm, Woman: 45 gm and Child: 33 gm.

Solution: Represent the above information by matrices in using matrix multiplication method

Solution: The members of the two families can be represented by the 2 × 3 matrix.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

And the recommended daily requirement of calories and proteins for each member can be represented by the 3 × 2 matrix:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
The total requirements of calories and proteins for each of the families is given by matrix multiplication
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
= ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Hence finally A requires 24,600 calories and 556 gm proteins and Family B requires 15,800 calories and 332 gm proteins.

Example 12: Three firms A, B and C supplied 40, 35 and 25 truckloads of stones and 5, 8 truckloads of sand respectively to a contractor. If the costs of stone and sand are ₹1200 and ₹500 per truck load respectively, find total amount paid by the contractor to each of these firms, buy using matrix method.

Solution: The amount of stone and sand supplied to a contractor by three firms A, B and C can represented by using matrix method.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
The cost per truck load of stone and stand can be represented by the column matrix.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Thus, the total amount paid by the contractor to each of these firms is given by the matrix product.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Hence the amount paid to firms A = ₹53,000; B = ₹44,500 and C = ₹ 34,000

DETERMINANTS

The determinant of a square matrix is a number which is associated with the square matrix. This number may be positive, negative or zero > the determinant of a square matrix A commonly denoted by det A or | A| or Δ. The matrices which are not square do not have determinants.

Determinants are quite useful to solving a system of linear equations. They are also equations. They are also helpful in expressing certain formulas.
The determinant of a (2 × 2) matrix A ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
The determinant of (3 × 3) matrix, A ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
And its defined as  ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
= a1 (b2c3-b3c2) – a2(b1c3-b3c1) + a3(b1c2-b2c1)

Minors and Cofactors of a Determinant:

Minor of the element of a determinant is the determinant of Mij by deleting ith row and jth column in which element is existing.

Cofactor of matrix A = (-1)i+j Mij , Where Mij = Minor of the element ith row and jth column.
Minors: Let us consider the determinant ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Cofactors

C11= (-1)1+1M11= M11, C12= (-1)1+2M12= -M21 ; C13= (-1)1+3M13= M13

C21= (-1)2+1M21= - M21 , C22= (-1)2+2M22= M22 ; C23= (-1)2+3M23= -M23

C31= (-1) 3+1M31= M31 , C32= (-1)3+2M32= -M32 ; C33= (-1)3+3M33= M33

The value of determinant can be defined in terms of cofactor matrix as

Δ = a11 c11 + a12 c12 + a13c13

or

Δ = a11 c11 + a21 c12 + a31 c13

Properties of Determinants:

1) The value of determinant remains unaltered interchanged if its rows or columns

interchanged.

2) The value of determinant change signs if any two rows (or columns) interchanges.

3) The value of determinant is zero if any two rows (any columns) then value of determinant

is equal to zero.

4) The value determinant becomes k times (where k is constant) if any row or columns

multiplied by k the value of determinant also multiplied by k.

5) The value of determinant is zero if any two rows (or column) are proportional then the

value of determinant is equal to zero.

6) If each element of any row (or column) is a sum of two numbers, the determinant can be

expressed as the sum of the determinants.

7) The value of determinant remains same if to any (or column) multiple of row (or column) is added or subtracted.

Singular and Non-Singular Matrices: Any Square Matrix A is singular, if | A | = 0 . The matrix is non-singular, if | A | ≠ 0 .
Example 13: If  ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation Prove that A is a singular matrix.
Solution: ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

A is singular matrix.

Example 14: Find the determinant value of the following matrices.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Solution by definition
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

= 1(45 – 48) –2(36 – 42) + 3(32 – 35)

= -3 + 12 - 9 = 0

Adjoint Matrix: Adjoint of A Matrix is the transpose of the Cofactor Matrix

Example 15: Find the Adjoint of the Matrix.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Solution:
The Co-factors of elements of ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation are calculated below
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

INVERSE OF A MATRIX
If A is a Square matrix and A ≠ 0 then
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Example 16:

Solve the following system of equations by matrix inversion method :

2x + 8y + 5z = 5

X + y + z = (-2)

X + 2y – z = -2

Solution: The given system of equations can be written in the form, AX = B.
Where (A) = ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
det (A)ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

= -6 + 16 + 5

= 15 ≠ 0

Hence, the system has a unique solution as A is non-singular. The solution is given by

X = A-1B

To find A-1, we find the cofactors.

A11 = -3; A12 = + 2; A13 = 1, A21 = + 18; A22 =-7; A23 + 4; A31 = 3; A22 = +3; A33 = -6
Co-factor of ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Adj: A = (co-factor A)T = ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Hence, x = -3; y = 2 and z = -1

Example 17: Show that the matrix ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundationsatisfies the equation

A2 –5A –2I = 0

Hence, deduce the value of A-1
Solution:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Since A2 – 5A – 2I = 0,

⇒A-1 (A2 – 5A – 2I) = 0

⇒ A-5I – 2A-1 = 0

⇒ 2A-1 = A – 5I
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Example 18:
Fine the inverse of the matrix.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Solution: Hence, solve the system of equations.

2x – 3y = 3

4x – 11y = 11
Hence, Inverse of ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
The given system of equations cab be written as

AX = B
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Therefore,
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Hence x = 0, y = -1
Example 19: Using the inverse of the coef cient matrix, solve the following system of equations:

x + y + z = 3
x + 2y + 3z =4
x + 4y + 9z = 6
Solution: The given equation can be written as AX = B. Where ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Since,
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

∴ The given equations are consistent, and the solution is given by X =A-1 B.

Let us find the cofactors:

A11 = 6; A12= -6; A13= 2; A21 = -5; A22 = 8; A23 = -3

A31 = 1; A32 = - 2 and A33 = 1
Then Cof.
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation


SOLUTION OF LINEAR EQUATIONS IN THREE VARIABLES (CRAMER’S RULE)

The solution of equations

a11 + a12y + a13z = d1

a21 + a22y + a23z = d2

a31 + a32y + a33z = d3
is give by ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
provided Δ≠0, and
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Notes : 1) If Δ = 0; and Δx = 0, Δy = 0 , Δz= 0; then the given equations will have in nite solutions and equations will be dependent.

2) If Δ = 0; and at least Δx , Δy = 0 , Δz; is not zero then the equations will have solution and the equations have no solution and the equations are said to be inconsistent.

Example 20: Solve the equations:

1) 2x – y + z = 4

X + 3y + 2z = 12

3x + 2y + 3z = 16

Solution: Considering the equations:

2x – y + z = 4

X + 3y + 2z = 12

3x + 2y + 3z = 16

By using Cramer’s Rule, the solution of the equations are given below:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Since Δ = 0; Δx= 0, Δy= 0 and Δz= 0,

There the equations are dependent and will have infinite solutions;

Example 21:

x + y + 3z = 6

x – 3y – 3z = –4

5x – 3y + 3z = 8
Solution: By applying Cramer’s rule, we get
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
Since Δx ≠ 0 and Δ = 0, therefore the system of equations are inconsistent.

Example 22: The given equations are:

x + y – z = –2

3x + 2y + 3z =13

2x + 7y + 4z = 31

Solution:
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

Δ = 1 (8 – 21) –1 (12 – 6) –1(21 – 4) = –13–6–17= –36

Δx = –2 (8 – 21) –1 (52 – 93) –1(91 – 62) = 26 + 41 – 29 = 38

Δy = 1 (52 – 93) +2 (12 – 6) –1(93 – 26) =–14 + 12 – 67= –96

Δz = 1 (62 – 91) –1 (93 – 26) –1(21 – 4) = –13 – 6 – 17 = –36
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation
ICAI Notes- Equations and Matrices- 2 | Quantitative Aptitude for CA Foundation

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FAQs on ICAI Notes- Equations and Matrices- 2 - Quantitative Aptitude for CA Foundation

1. What are the basic operations that can be performed on matrices?
Ans. The basic operations that can be performed on matrices include addition, subtraction, multiplication, and scalar multiplication.
2. How do you determine the order of a matrix?
Ans. The order of a matrix is determined by the number of rows and columns it has. For example, a matrix with 3 rows and 4 columns is said to have an order of 3x4.
3. How are equations solved using matrices?
Ans. Equations can be solved using matrices by representing the coefficients of the variables as a matrix and the constants on the right-hand side as another matrix. By performing matrix operations, such as Gaussian elimination or inverse matrix method, the values of the variables can be obtained.
4. What is the determinant of a matrix?
Ans. The determinant of a square matrix is a scalar value that can be calculated using a specific formula. It provides information about the invertibility of the matrix and plays a crucial role in solving systems of linear equations.
5. How does matrix multiplication work?
Ans. Matrix multiplication involves multiplying the elements of one matrix with the corresponding elements of another matrix and summing them up. The resulting matrix has dimensions determined by the number of rows of the first matrix and the number of columns of the second matrix.
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