[Maths Class Notes] on Determinants and Matrices Pdf for Exam

Matrices and determinants are important topics for class 12th board exams, JEE, and various other competitive examinations. Our matrices and determinants notes and solved examples will help you grasp the fundamental ideas related to this chapter such as types of matrices and the definition of the determinant. The primary question that arises is what is a matrix? An ordered rectangular array of numbers, functions, symbols, or objects is called a matrix. A matrix has the order m×n if it has m rows and n columns.

Such an m×n matrix are represented as:

[begin{bmatrix}a1 & cdots  & an  \vdots  & ddots  & vdots  \am & cdots  & amn \ end{bmatrix}]

Types of Matrices

• Diagonal matrix – A square matrix whose all elements except those in the main diagonal are zero -[begin{bmatrix} 2 & 0 \ 0 & 3 \ end{bmatrix}], [begin{bmatrix} 1 & 0 & 0 \ 0 & 6 & 0 \ 0 & 0 & 3 \ end{bmatrix}]

• Scalar matrix – A square matrix whose diagonal elements are all equal and all elements except those in the main diagonal are zero – [begin{bmatrix} 3 & 0 \ 0 & 3 \ end{bmatrix}], [begin{bmatrix} 7 & 0 & 0 \ 0 & 7 & 0 \ 0 & 0 & 7 \ end{bmatrix}]

• Identity matrix – A square matrix whose main diagonal elements are ‘1’ and the other elements are all zero. It is denoted by ‘I’- [begin{bmatrix} 1 & 0 \ 0 & 1 \ end{bmatrix}], [begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ end{bmatrix}]

• Null matrix – A matrix of any order, all of whose elements are zero- [begin{bmatrix} 0 & 0 \ 0 & 0 \ end{bmatrix}], [begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \ end{bmatrix}]

• Triangular matrix – It is a square matrix in which all the elements above or below the main diagonal are zero – [begin{bmatrix} 1 & 0 & 0 \ 7 & 1 & 0 \ 3 & 4 & 3 \ end{bmatrix}], [begin{bmatrix} 1 & 3 & 1 \ 0 & 3 & 3 \ 0 & 0 & 2 \ end{bmatrix}]

• Transpose of a matrix – The matrix obtained by interchanging the rows and columns of a matrix A and is denoted by [A^{T}] −if A=[begin{bmatrix} 1 & 2 & 0 \ 6 & 1 & 7 \ 5 & 0 & 3 \ end{bmatrix}] its transpose [A^{T}] = [begin{bmatrix} 1 & 6 & 5 \ 2 & 1 & 0 \ 0 & 7 & 3 \ end{bmatrix}]

Inverse of a matrix usually applies for square matrices, and there exists an inverse matrix for every m×n square matrix. If the square matrix is represented by A, then its inverse is denoted by A-1, and it satisfies the property, 

[AA^{-1}]= [A^{-1}]A = I, where I is the identity matrix. 

The determinant of the square matrix should be non-zero.

Operations on Matrices

The following operations can be performed between two or more matrices:

Solved Examples on Matrix Operations:

Addition of Matrix 

1. If A=[begin{bmatrix} 2 & 5 & -1 \ 4 & 1 & 3 \ end{bmatrix}] and B =[begin{bmatrix} 3 & 2 & 4 \ -1 & 5 & 2 \ end{bmatrix}], 

then A + B  = [begin{bmatrix} 2 & 5 & -1 \ -1 & 5 & 2 \ end{bmatrix}] +[begin{bmatrix} 3 & 2 & 4 \ -1 & 5 & 2 \ end{bmatrix}] 

=[begin{bmatrix} 2+3 & 5+2 & -1+4 \ 4-1 & 1+5 & 3+2 \ end{bmatrix}] 

=[begin{bmatrix} 5 & 7 & 3 \ 3 & 6 & 5 \ end{bmatrix}] 

Subtraction of Matrix 

2. If A = [begin{bmatrix} 1 & 3 & -2 \ 4 & 7 & 5 \ end{bmatrix}]

and B =[begin{bmatrix} -2 & 1 & -1 \ 3 & 5 & 2 \ end{bmatrix}] 

then A-B = A+(-B) =[begin{bmatrix} 1 & 3 & -2 \ 4 & 7 & 5 \ end{bmatrix}] + [begin{bmatrix} 2 & -1 & 1 \ -3 & -5 & -2 \ end{bmatrix}]

= [begin{bmatrix} 1+2 & 3-1 & -2+1 \ 4-3 & 7-5 & 5-2 \ end{bmatrix}]

= [begin{bmatrix} 3 & 2 & -1 \ 1 & 2 & 3 \ end{bmatrix}]

Multiplication of Matrix

3. If A=[begin{bmatrix} 1 & 1 \ 0 & 2 \ 1 & 1 \ end{bmatrix}] and B  = [begin{bmatrix} 1 & 2 \ 2 & 2 \ end{bmatrix}], 

Then 

AB = [begin{bmatrix} 1 & 1 \ 0 & 2 \ 1 & 1 \ end{bmatrix}] [begin{bmatrix} 1 & 2 \ 2 & 2 \ end{bmatrix}]

=[begin{bmatrix} 1times 1+1times 2 & 1times 2+1times 2 \ 0times 1+2times 2 & 0times 2+2times 2 \ 1times 1+1times 2 & 1times 2+1times 2 \ end{bmatrix}] [begin{bmatrix} 3 & 4 \ 4 & 4 \ 3 & 4 \ end{bmatrix}]

Determinant of a Matrix

Next, we will learn the definition of the determinant of a matrix. The determinant of a matrix is defined as a scalar value that can be calculated from the elements of a square matrix. It encodes some of the properties of the linear transformation that the matrix describes and is indicated as det A, det (A), or |A|. Let us clarify it further:

A square matrix A of a specific order has a number associated with it, and this number is called the determinant of matrix A. In order for a determinant to be associated with a matrix, the latter has to be a square matrix. 

Thus, for the 2×2

square matrix A=[begin{bmatrix} a1 & b1 \ a2 & b2 \ end{bmatrix}]

the symbol |A|=[begin{bmatrix} a1 & b1 \ a2 & b2 \ end{bmatrix}]

signifies a determinant of second- order. 

Its value is defined as: [begin{bmatrix} a1 & b1 \ a2 & b2 \ end{bmatrix}]= a1b2−a2b1

Similarly, for a 3 x 3 square matrix A=[begin{bmatrix} a1 & b1 & c1 \ a2 & b2 & c2 \ a3 & b3 & c3 \ end{bmatrix}]

the symbol |A|=[begin{bmatrix} a1 & b1 & c1 \ a2 & b2 & c2 \ a3 & b3 & c3 \ end{bmatrix}]

signifies a determinant of third-order. 

Its value is defined as:

|A|=a1[begin{vmatrix} b2 & c2 \ b3 & c3 \ end{vmatrix}] −b1[begin{vmatrix} a2 & c2 \ b3 & c3 \ end{vmatrix}] +c1[begin{vmatrix} a2 & b2 \ a3 & b3 \ end{vmatrix}]

Solved Example:

1. Find the determinant of matrix A, if A=[begin{bmatrix} 2 & 5 \ 1 & 3 \ end{bmatrix}] 

Solution: |A|= [begin{bmatrix} 2 & 5 \ 1 & 3 \ end{bmatrix}] 

(2 x 3) – (5 x 1) = 6 – 5 = 1

Properties of Determinant

Having introduced the determinant definition in math, let us go through some of the properties of determinant:

If all the elements of a matrix are zero, then the determinant of the matrix is zero.

For an identity matrix I of the order m×n, determinant of I, |I|= 1.

If the matrix A has a transpose [A^{T}], then |[A^{T}]| = |A|.

If A-1 is the inverse of matrix A, then |[A^{-1}]| = 1/|A| = [|A|^{-1}]

If two square matrices A and B are of the same size, then |AB| = |A| |B|.

If c is a constant and a matrix A has the size b×b, then |cA| = [c^{b}] |A|.

For triangular matrices, the product of the diagonal elements gives the determinant of the matrix.

Laplace’s formula: Using this formula, the determinant of a matrix is expressed in terms of its minors. If the matrix Nxy  is the minor of matrix M, obtained by eliminating the xth and yth column and has a size of ( j-1 × j-1), then the determinant of the matrix M will be given as:

|M| = [sum_{y=1}^{i} (−1)^{x+y} a_{x,y} N_{x,y}] where [(−1)^{x+y}] [N_{x,y}] is the cofactor.

Adjugate matrix – It is determined by transposing the matrix that contains the cofactors and is calculated using the equation:

[(Adj (M))_{x,y} = (-1)^{x+y} N_{x,y}]

Significance of Matrices and Determinants in Mathematics

Matrices and determinants are used to calculate linear equations in two or three variables. Matrices and determinants are also used to determine if a system is stable or not. The determinant can be used to solve linear equations, to capture how linear transformations alter area or volume, and to modify variables in numerical methods. The determinant can be thought of as a function with a square matrix as its source and an integer as its result. A determinant is a significant number that is given to any square matrix and has geometrical and mathematical significance and hence they make an important topic to study in mathematics for real-life applications as well as for the exam point of view. 

Significance of Matrices in Business Administration

A decision matrix can assist you in not just making complex judgments, but also in prioritizing activities, solving problems, and crafting reasons to support a previous decision. It’s an excellent decision-making technique if you’re deciding between a few similar options based on a variety of quantitative parameters.

The value of the determinant of the matrix is always positive.

A square matrix’s determinant can be found by applying row functions to obtain a triangular version for the matrix which means, all inputs either above or below the axis are 0. The determinant is the result of the elements on the triangle form’s principal axis.

History of Determinants and Matrices

The origins of matrices and determinants can be traced all the way back to the 2nd century BC, with hints dating to the mid-4th century BC. Nevertheless, it was not before the latter half of the 17th century that the concepts resurfaced and true progress began. It’s not unexpected that the research of processes of linear equations led to the development of matrices and determinants. The Babylonians researched issues that resulted in concurrent linear equations, and some of these have been recorded on clay tablets.

Conclusion 

Here, we have provided concepts and Solutions to questions of topic Determinants and Matrices in an elaborate manner. You can find everything you’re looking  for in one place. In the PDFs, which are also downloadable for free, students can go through concepts, Definitions, and questions carefully and understand the concepts used to solve these questions. This will help the students immensely in their examinations.