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Applied Mathematics · Class 12 Commerce

Ch 2Algebra — Class 12 Applied Mathematics, concept-first.

This concept map shows how the chapter fits together. Algebra here splits into three areas: Matrices (their order, types, algebra, transpose, symmetric/skew forms and inverse), Determinants (minors, cofactors, adjoint, area of a triangle and their properties), and using both to solve systems of linear equations (invers…

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Matrix Representation Order

Imagine you're a teacher taking attendance. You have a list of 5 students and you call out their names one by one. The order matters — "Ravi, Priya, Anjali, Vikram, Sneha" is a specific sequence.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Concept Map

This concept map shows how the chapter fits together. Algebra here splits into three areas: Matrices (their order, types, algebra, transpose, symmetric/skew forms and inverse), Determinants (minors, c…

2.2

Introduction

Before diving into formulas, it helps to see where matrices show up in everyday life. Think of the precise rows and columns of soldiers in a Republic Day parade, students seated by row and column in a…

2.3

Matrix

11 Q

A matrix is a rectangular array of numbers, symbols or expressions arranged in rows and columns, enclosed in a single pair of brackets.

+Worked Examplesi4 questions
  1. Example 1Shalabh has 3 books, 2 pens and 3 notebooks while Rashmi has 1 pen, 4 books and 5 notebooks in their respective school bags. Express the inf…Free
  2. Example 2Write the coordinates of triangle ABC with vertices $A(4,-1)$, $B(3,2)$ and $C(2,-4)$ in a matrix formation.Free
  3. Example 3If a matrix has 4 elements, what are the possible orders such a matrix can have?Preview
  4. Example 4Construct a $3\times 3$ matrix whose elements are given by $a_{ij} = \dfrac{i+2j}{5}$.Preview
+Exercise Ai7 questions
  1. Q1Identify the type of matrices given below and write the order of each matrix: (i) $A = \begin{bmatrix} 2 & 3 \end{bmatrix}$ (ii) $B = \begin…Free
  2. Q2(i) $A = \begin{bmatrix} 0 & -4 & 3 \\ 1 & 0 & -7 \\ 2 & 2 & 0 \end{bmatrix}$, write the element $a_{12}$. (ii) $B = \begin{bmatrix} -9 & 4…Free
  3. Q3Construct matrix $A = [a_{ij}]$ of order $2\times 3$ where $a_{ij} = \dfrac{(i+2j)^2}{2}$.Free
  4. Q4Construct matrix $B = [b_{ij}]$ of order $2\times 2$ where $b_{ij} = \dfrac{|i-j|}{3}$.Preview
  5. Q5How many distinct $2\times 2$ matrices can be formed by using numbers 5, 7 and $-1$? Justify your answer.Preview
  6. Q6A matrix has 14 elements. How many matrices of different orders are possible?Preview
  7. Q7Find the values of $a$, $b$, $c$ and $d$ from the equation: $\begin{bmatrix} 14 & a+b \\ c+d & b+c \end{bmatrix} = \begin{bmatrix} a & -b \\…Preview
2.3.1

Types of Matrices

Matrices come in several standard shapes, each with its own name and use.

+Worked Examplesi2 questions
  1. Example 5If $A = \begin{bmatrix} 1 & 2a \\ -8 & b+1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -6 \\ -8 & 13 \end{bmatrix}$ are equal matrices, find…Free
  2. Example 6Find the values of $a$, $b$, $c$ and $d$ from the following equation: $\begin{bmatrix} 4 & 24 \\ -3 & 11 \end{bmatrix} = \begin{bmatrix} 2a+…Preview
2.4

Algebra of Matrices

With the vocabulary of matrices in place, we can now define arithmetic on them, mirroring what you already know for ordinary numbers.

2.4.1

Multiplication of a Matrix by a Scalar Value

Multiplying a matrix by a scalar (an ordinary number) is the simplest matrix operation: every single element of the matrix gets multiplied by that number.

2.4.2

Addition of Matrices

Two matrices can be added only when they have exactly the same order — you cannot add a matrix to a matrix, since there is no sensible way to pair up their elements.

2.4.3

Subtraction of Matrices

Subtraction works exactly like addition, just with elements subtracted instead of added. For two matrices and of the same order , their difference is

+Worked Examplesi3 questions
  1. Example 7For the matrices $A = \begin{bmatrix} 3 & 4 & 0 \\ -1 & 12 & 3 \\ 6 & 1 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 7 & 7 & 2 \\ -11 & 0 & 2 \\…Free
  2. Example 8If $X = \begin{bmatrix} -1 & 3 \\ 8 & 4 \end{bmatrix}$ and $Y = \begin{bmatrix} -5 & 1 \\ -1 & -2 \end{bmatrix}$ then find the matrix $Z$, s…Preview
  3. Example 9Find $X$ and $Y$ if $X+Y = \begin{bmatrix} -1 & 13 \\ 2 & 4 \end{bmatrix}$ and $X-Y = \begin{bmatrix} 5 & -8 \\ 3 & 0 \end{bmatrix}$.Preview
2.4.4

Multiplication of Matrices

Multiplying two matrices is less straightforward than adding or subtracting them, because it isn't simply a matter of multiplying corresponding elements.

+Worked Examplesi4 questions
  1. Example 10Let $A = \begin{bmatrix} 3 & 5 \\ -4 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} -9 & 2 \\ 1 & -7 \end{bmatrix}$, find $AB$ and $BA$.Free
  2. Example 11Find $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}\begin{bmatrix} -2 & 0 & 1 \\ 9 & 3 & -3 \\ 6 & 1 & -2 \end{bmatri…Free
  3. Example 12For $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$, prove that $A^2-4A+I = O$, where $O$ is a zero matrix.Preview
  4. Example 13Given that $X_{2\times n}$, $Y_{3\times k}$, $Z_{2\times p}$, $W_{n\times 3}$ and $P_{p\times k}$ are matrices of specified order. What are…Preview
2.5

Special Matrices

Beyond the basic shapes already introduced, a few special matrices — built from an existing matrix rather than defined from scratch — turn out to be especially useful: the transpose of a matrix, obtai…

+Exercise Bi10 questions
  1. Q1Complete the following table (Order of the matrix): | A | B | A±B | AB | |---|---|---|---| | $2\times 2$ | $2\times 2$ | | | | $2\times 3$ |…Free
  2. Q2For $A = \begin{bmatrix} 6 & -5 \\ -7 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 1 & -3 \\ -2 & 4 \end{bmatrix}$ and $C = \begin{bmatrix} -2 &…Free
  3. Q3Consider $A = \begin{bmatrix} 1 & 3 & 4 \\ -2 & 1 & 2 \\ 3 & -2 & 1 \end{bmatrix}$, verify that $A.I = I.A = A$, where $I$ is the identity m…Free
  4. Q4If $A = \begin{bmatrix} 1 & -3 \\ -2 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -4 \\ -1 & 3 \end{bmatrix}$ then show that (i) $(A+B)'…Preview
  5. Q5Do as directed: (i) For $A = \begin{bmatrix} 6 & -5 \\ -7 & 4 \end{bmatrix}$, find $A^2 - 6A$. (ii) Evaluate $\begin{bmatrix} 2 & 1 & 3 \end…Preview
  6. Q6If $A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}$, then find…Preview
  7. Q7Given $A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5…Preview
  8. Q8For $A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{bmatrix}$ show that $A^3 - 23A - 40I = O$, where $I$ is an identity matri…Preview
  9. Q9Two booksellers A and B sell the textbook of Mathematics and Applied Mathematics. In the month of March, bookseller A sold 250 books of Math…Preview
  10. Q10Cost of a pen and a notebook are ₹12 and ₹27 respectively. On a given day, shopkeeper P sells 5 pens and 7 notebooks, whereas shopkeeper Q s…Preview
2.5.1

Transpose of a Matrix

The transpose of a matrix is the matrix obtained by interchanging its rows and columns — the first row of becomes the first column of the transpose, the second row becomes the second column, and so on…

+Worked Examplesi1 question
  1. Example 14If $A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 3 \\ 5 & -4 \end{bmatrix}$ then find the matrix $A'-2B$…Preview
2.5.2

Symmetric and Skew Symmetric Matrices

A square matrix is called symmetric when it is unchanged by transposition, that is — every element mirrored across the main diagonal is identical to the entry on the other side, for all .

+Worked Examplesi3 questions
  1. Example 15If $P = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix}$, then verify that $P'P = I$, where $I$ is an identity matrix.Free
  2. Example 16If $A$, $B$ are symmetric matrices of same order, then what can be said for matrix $AB - BA$?Preview
  3. Example 17Express the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & -1 & 0 \\ 3 & 5 & 1 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric…Preview
2.6

Determinant

For a pair of simultaneous linear equations and , you already know the system has a unique solution precisely when .

+Exercise Ci7 questions
  1. Q1Evaluate the following: (i) $\begin{vmatrix} 4 & -2 \\ 6 & -3 \end{vmatrix}$ (ii) $\begin{vmatrix} 7 & 1 \\ 4 & -7 \end{vmatrix}$ (iii) $\be…Free
  2. Q2Find the area of the triangle with vertices $(-2,-3)$, $(-1,-8)$ and $(3,2)$.Free
  3. Q3For what value of "$k$" the points $(k,7)$, $(-4,5)$ and $(1,-5)$ are collinear.Free
  4. Q4Represent the given matrices as the sum of a symmetric and skew symmetric matrices: (i) $\begin{bmatrix} 3 & 1 \\ -1 & 8 \end{bmatrix}$ (ii)…Preview
  5. Q5Evaluate using properties of determinants: (i) $\begin{vmatrix} b-c & c-a & a-b \\ c-a & a-b & b-c \\ a-b & b-c & c-a \end{vmatrix}$ (ii) $\…Preview
  6. Q6Prove the following using properties of determinants: (i) $\begin{vmatrix} -a^2 & ab & ac \\ ab & -b^2 & bc \\ ac & bc & -c^2 \end{vmatrix}…Preview
  7. Q7Find adjoint $A$ if: (i) $A = \begin{bmatrix} 2 & 1 \\ -3 & 5 \end{bmatrix}$ (ii) $A = \begin{bmatrix} -52 & 11 \\ 0 & 51 \end{bmatrix}$ (ii…Preview
2.6.1

Minor of an Element

The minor of an element in a determinant is found by deleting the entire row and the entire column that the element sits in, and taking the determinant of whatever remains.

+Worked Examplesi4 questions
  1. Example 18Find the minor of element $-11$ in $A = \begin{vmatrix} 6 & -11 \\ -1 & 3 \end{vmatrix}$.Free
  2. Example 19Given $A = \begin{bmatrix} 3 & 5 \\ -4 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} -9 & 2 \\ 1 & -7 \end{bmatrix}$, find the following (i) $…Free
  3. Example 20Find $x$ if $\begin{vmatrix} 3 & -6 \\ 4 & 0 \end{vmatrix} = \begin{vmatrix} 3 & x^2 \\ x & -1 \end{vmatrix}$.Preview
  4. Example 21Find the minor of element $7$ in the determinant $B = \begin{vmatrix} 2 & 3 & 0 \\ -3 & 1 & 7 \\ 1 & -2 & 5 \end{vmatrix}$.Preview
2.6.2

Cofactor of an Element of a Determinant

The cofactor of an element refines its minor with a sign that depends on position. For the element in row , column , the cofactor is denoted (or ) and defined as

+Worked Examplesi2 questions
  1. Example 22Evaluate $\Delta = \begin{vmatrix} 0 & \cos x & -\sin x \\ -\cos x & 0 & \cos y \\ \sin x & -\cos y & 0 \end{vmatrix}$.Free
  2. Example 23Find minors and cofactors of the elements of the determinant $A = \begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$ and c…Preview
2.6.3

Adjoint of a Matrix

The adjoint of a square matrix , written , is formed by first replacing every element of with its own cofactor (giving the cofactor matrix), and then taking the transpose of that cofactor matrix.

+Worked Examplesi1 question
  1. Example 24Find $k$, if $A = \begin{bmatrix} -2 & 3 \\ k & 4 \end{bmatrix}$ is a singular matrix.Preview
2.6.4

Area of Triangle

The determinant also gives a compact formula for the area of a triangle whose vertices are known coordinates. For a triangle with vertices , and ,

+Worked Examplesi2 questions
  1. Example 25Find the area of the triangle with vertices $A(5,4)$, $B(2,-6)$ and $C(-2,4)$.Free
  2. Example 26For what value of $k$, points $P(3,-2)$, $Q(8,8)$ and $R(k,2)$ are collinear.Preview
2.6.5

Properties of a Determinant

Several structural properties make determinants easier to compute and manipulate, without ever multiplying everything out by hand.

+Worked Examplesi6 questions
  1. Example 27Evaluate $\Delta = \begin{vmatrix} 42 & 2 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{vmatrix}$.Free
  2. Example 28Evaluate $\Delta = \begin{vmatrix} 1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b \end{vmatrix}$.Free
  3. Example 29Given that $a$, $b$ and $c$ are in A.P., evaluate $\Delta = \begin{vmatrix} 2y+4 & 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c…Preview
  4. Example 30Without expanding prove that $\Delta = \begin{vmatrix} x & y & z \\ x^2 & y^2 & z^2 \\ x^3 & y^3 & z^3 \end{vmatrix} = xyz(x-y)(y-z)(z-x)$.Preview
  5. Example 31Without expanding, evaluate $\Delta = \begin{vmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{vmatrix} = 2abc…Preview
  6. Example 32Evaluate without expanding $\Delta = \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = abc\left(1+\dfrac{1}{a}+\dfra…Preview
2.7

Inverse of a Matrix

A square matrix of order is called invertible if there exists another square matrix of the same order such that , where is the identity matrix of order .

2.7.1

Finding Inverse Matrix by Elementary Operations (Transformation)

An elementary operation (or transformation) is one of three simple moves you can make on the rows (or, separately, the columns) of a matrix: interchanging two rows, denoted ; multiplying every element…

+Worked Examplesi3 questions
  1. Example 33By using elementary row transformations, find inverse of matrix $A = \begin{bmatrix} 1 & 0 \\ -3 & 5 \end{bmatrix}$.Free
  2. Example 34Find the inverse of matrix $A$ using elementary row operations where $A = \begin{bmatrix} 2 & 3 & 10 \\ 4 & -6 & 5 \\ 6 & 9 & -20 \end{bmatr…Preview
  3. Example 35Find $A^{-1}$, if $A = \begin{bmatrix} 10 & -2 \\ -5 & 1 \end{bmatrix}$.Preview
2.7.2

Finding Inverse Matrix by Inverse of Coefficient Matrix Method

Recall that for any square matrix of order , the adjoint satisfies . Dividing through by , whenever that's possible, immediately gives a ready-made formula for the inverse:

+Worked Examplesi1 question
  1. Example 36Find the inverse of matrix $A$ using inverse of the coefficient matrix method when $A = \begin{bmatrix} 2 & 3 & 10 \\ 4 & -6 & 5 \\ 6 & 9 &…Preview
2.8

Solving System of Simultaneous Linear Equations

A system of simultaneous linear equations in two or three variables can be rewritten compactly as a single matrix equation, , where is the matrix of coefficients, is the column of unknowns, and is the…

+Exercise Di4 questions
  1. Q1Find the inverse of the given matrices, by using elementary transformations: (i) $\begin{bmatrix} -5 & -1 \\ 3 & 2 \end{bmatrix}$ (ii) $\beg…Free
  2. Q2Find inverse of the given matrices, by using Adjugate (matrix) method: (i) $\begin{bmatrix} 4 & -1 \\ 3 & 2 \end{bmatrix}$ (ii) $\begin{bmat…Free
  3. Q3Solve the following system of equations by (i) Matrix method (ii) Row reduction method: (a) $2x - 3y = -4$, $3x + 5y = 13$ (b) $x + y = 1$,…Preview
  4. Q4Solve the following system of equation using Cramer's rule. (i) $2x - 3y = -4$, $3x + 5y = 13$ (ii) $x + y = 1$, $5x - 7y = 29$ (iii) $5x -…Preview
2.8.1

Inverse of Coefficient Matrix

Once a system of equations is written in matrix form , and provided the coefficient matrix is invertible (that is, ), the solution can be found directly by multiplying both sides on the left by :

+Worked Examplesi3 questions
  1. Example 37Solve the following system of equation finding the inverse of coefficient matrix: $x - y = 5$; $2x + 3y = -1$.Free
  2. Example 38Solve for $x$, $y$ and $z$: $2x - 3y = 5$; $5x + 3y = 2$.Preview
  3. Example 39Solve the following system of equation using matrix method: $x + y + z = 10$, $2x + y = 13$, $x + y - 4z = 0$.Preview
2.8.2

Cramer's Rule

Cramer's rule solves a system of linear equations using only determinants, without ever explicitly computing an inverse.

+Worked Examplesi4 questions
  1. Example 40Solve the following system of equation using Cramer's rule: $2x - 3y = 5$; $5x + 3y = 2$.Free
  2. Example 41Solve the following system of equations using Cramer's rule: $x + y + z = 10$, $2x + y = 13$, $x + y - 4z = 0$.Free
  3. Example 42Solve the following system of equations using Cramer's rule: $2x - 3y = 5$; $-4x + 6y = -10$.Preview
  4. Example 43Solve the following system of equations using Cramer's rule: $x - 2y + 3z = 1$, $2x + y - z = 3$ and $3x - y + 2z = -2$.Preview
2.8.3

Row Reduction Method

The row reduction method solves a system by writing the coefficients and constants together as a single augmented matrix , and then applying the same elementary row operations used earlier to find inv…

+Worked Examplesi4 questions
  1. Example 44Solve using row reduction method: $x - 3y = 9$, $2x + y + 3 = 0$.Free
  2. Example 45Use row reduction method to solve the given system of equations: $3x + 2y - z = 1$, $x + 2y - 2z = 0$ and $2x + y - 3z = -1$.Free
  3. Example 46Three shopkeepers A, B and C are using polythene bags, handmade bags and newspaper bags. A uses 20, 30 and 40 number of bags of respective t…Preview
  4. Example 47A school plans to award ₹6000 in total to its students to reward for certain values - honesty, regularity and hard work. When three times th…Preview
2.9

Application of Matrices and Determinants

Matrices and determinants aren't just an abstract exercise — they show up as working tools across a surprising range of fields.

+Exercise Ei4 questions
  1. Q1Solve the following problem using Leontief input-output model. | | Sector 1 | Sector 2 | Total | |---|---|---|---| | Sector 1 | 2 | 5 | 10 |…Free
  2. Q2Solve the following problem using Leontief input-output model. | | FI | AI | Total | |---|---|---|---| | Food industry | 20 | 10 | 40 | | Ag…Free
  3. Q3Solve the following problem using Leontief input-output model. | | Sector 1 | Sector 2 | Total | |---|---|---|---| | Sector 1 | 12 | 20 | 40…Preview
  4. Q4Solve the following problem using Leontief input-output model. | | Sector 1 | Sector 2 | Total | |---|---|---|---| | Sector 1 | 5 | 7 | 30 |…Preview
2.9.1

Leontief Input-output Model for Two Variables

In a real economy, sectors rarely work in isolation — an automobile sector might rely on steel, electricity and rubber, while those very sectors also consume some of what the automobile sector produce…

+Worked Examplesi1 question
  1. Example 48Prepare an input-output table for Transport industry (TI) and Food industry (FI). Food industry produces 50 units. Out of these 20 units con…Preview
2.10

Unit Summary

This section pulls together the core definitions, properties and formulae of the Algebra unit of CBSE Class 12 Applied Mathematics (subject code 241) — matrices, their operations, determinants and the…