JEE

# JEE Chapter - Matrices

Get plus subscription and access unlimited live and recorded courses

## Introduction to Matrices

Matrices are one of the most exciting, simple, and important concepts in Mathematics. This chapter is totally new from the student's perspective, as it appears right after the 11th. As a result, some students may find Matrices difficult to comprehend and solve problems with at first. However, as you solve more questions in this chapter and become more comfortable with the concepts and the chapter as a whole, you will discover that this is one of the easier chapters.

Matrix, types of matrices, operations on matrices, transpose of a matrix, application of matrices, some of the solved examples, and some previous year questions will be discussed in this article.

### Important Topics of Matrices

• Matrices

• Operation on Matrices

• Types of Matrices

• Transpose of Matrix

• Matrix Multiplication

• Inverse Matrix

• Inverse of 3 by 3 Matrix

### What is the Matrix?

In Mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. When writing matrices, box brackets are usually applied. The horizontal and vertical lines of entries in a matrix are referred to as rows and columns, respectively. The number of rows and columns in a matrix determines the size of the matrix. A matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, while m and n are called its dimensions. The dimensions of the following matrix are 2 x 3 up(read “two by three”), because there are two rows and three columns.

$A={\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}$

Image: Matrix dimension - Each element of a matrix is often denoted by a variable with two subscripts.

The elements or entries of a matrix are the individual items (numbers, symbols, or phrases) that make up the matrix.

Two matrices can be added or removed element by element as long as they are the same size (have the same number of rows and columns). The rule for matrix multiplication, on the other hand, is that two matrices can only be multiplied if the first's number of columns equals the second's number of rows. A scalar from the related field can be multiplied element-by-element by any matrix.

Row vectors are matrices with a single row, and column matrices are matrices with a single column. A square matrix is a matrix with the same number of rows and columns. A matrix with no rows or columns, known as an empty matrix, is beneficial in particular situations, such as computer algebra applications.

### Types of Matrix

• Row Matrix - A matrix having one row is called a row matrix.

• Column Matrix - A matrix having one column is called a column matrix.

• Square Matrix - A matrix of order m×n is called a square matrix if m = n.

• Zero Matrix - A = [aij] m×n is called a zero matrix if aij = 0 for all i and j.

• Upper triangular Matrix - A = [aij]m×n is said to be upper triangular if aij= 0 for i > j.

• Lower triangular Matrix - A = [aij]m×n is said to be lower triangular if aij = 0 for i < j.

• Diagonal Matrix - A square matrix [aij]m×n is said to be diagonal, if aij = 0 for i ≠ j.

• Scalar Matrix - A diagonal matrix A = [aij]m×n is said to be scalar, if aij = k for i = j.

• Unit Matrix (Identity matrix) - A diagonal matrix A = [aij]n is a unit matrix if aij = 1 for i = j.

• Comparable Matrices - Two matrices A and B are said to be comparable if they have the same order.

### Matrix Addition and Subtraction

Matrixes are used to represent systems or to list data. We can execute operations on matrices since the elements are integers. By adding/deleting corresponding entries, we can add or subtract the given matrices.

The entries must match in order to accomplish this. As a result, adding and subtracting matrices is only conceivable when their dimensions are the same. Because matrix addition is both commutative and associative, the following holds true:

$\displaystyle A+B=B+A$

$\displaystyle (A+B)+C=A+(B+C)$

Adding matrices is a simple process. Simply multiply each element in the first matrix by the number in the second matrix.

$\displaystyle \begin {pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}+\begin{pmatrix} 10 & 20 & 30 \\ 40 & 50 & 60 \end{pmatrix}=\begin {pmatrix} 11 & 22 & 33 \\ 44 & 55 & 66 \end {pmatrix}$

Note that the element  in the first matrix, 1, adds to the element X11 in the second matrix, 10, to produce element X11 in the resultant matrix, 11. Also note that both matrices being added are 2 x 3, and the resulting matrix is also 2 x 3.

And, the 2 matrices of the different dimensions cannot be added.

As you know, the subtracting works much the same way except that you subtract instead of adding.

$\displaystyle \begin{pmatrix} 10 & -20 & 30 \\ 40 & 50 & 60 \end{pmatrix}-\begin{pmatrix} 1 & -2 & 3 \\ 4 & -5 & 6 \end{pmatrix}=\begin{pmatrix} 9 & -18 & 27 \\ 36 & 55 & 54 \end{pmatrix}$

It's worth repeating that the new matrix has the same dimensions as the originals and that you can't subtract two matrices of different dimensions. When subtracting signed numbers, be cautious.

### Matrix Multiplication

1. Scalar Multiplication

Scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction in an intuitive geometrical context. What does it mean to multiply a number by 3? It means you add the number to itself 3 times. Multiplying a matrix by 3 means the same thing; you add the matrix to itself 3 times or simply multiply each element by that constant.

$\displaystyle 3\cdot \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}=\begin{pmatrix} 3 & 6 & 9 \\ 12 & 15 & 18 \end{pmatrix}$

The resulting matrix has the same dimensions as the original and the Scalar multiplication has the following properties:

• Left and right distributivity: $(c+d)\textbf{M} = \textbf{M}(c+d) = \textbf{M}c+\textbf{M}d$

• Associativity: $(cd)\textbf{M} = c(d\textbf{M})$

• Identity: $1\textbf{M} = \textbf{M}$

• Null: $0\textbf{M} = \textbf{0}$

• Additive inverse: $(-1)\textbf{M} = -\textbf{M}$

2. Multiplication with Another Matrix

When multiplying matrices, the elements of the first matrix's rows are multiplied by the elements of the second matrix's columns.

If A is an n x m matrix and B is an m x p matrix, the result AB of their multiplication is an n x p matrix defined only if the number of columns m in A is equal to the number of rows m in B.  Before multiplying the matrices, double-check that this is the case, as there would be "no solution" otherwise.

### General Definition and Process: Matrix Multiplication

Matrix multiplication is simply multiplying every element of each row of the first matrix times every element of each column in the second matrix, whereas scalar multiplication is simply multiplying a value through all the elements of a matrix. Although scalar multiplication is significantly easier than matrix multiplication, there is a pattern.

When multiplying matrices, the elements of the first matrix's rows are multiplied by the elements of the second matrix's columns. The resultant matrix's entries are computed one by one.

For 2 matrices, the final position of the product obtained is shown below:

$\displaystyle \begin{bmatrix} { a }_{ 11 } & { a }_{ 12 } \\ \cdot & \cdot \\ { a }_{ 31 } & { a }_{ 32 } \\ \cdot & \cdot \end{bmatrix}\begin{bmatrix} \cdot & { b }_{ 12 } & { b }_{ 13 } \\ \cdot & { b }_{ 22 } & { b }_{ 23 } \end{bmatrix}=\begin{bmatrix} \cdot & x_{ 12 } & \cdot \\ \cdot & \cdot & \cdot \\ \cdot & \cdot & { x }_{ 33 } \\ \cdot & \cdot & \cdot \end{bmatrix}$

The values at the intersections marked with circle are:

$\displaystyle {x}_{12}=({a}_{11},{a}_{12}) \cdot ({b}_{12},{b}_{22})=({a}_{11} {b}_{12}) +({a}_{12} {b}_{22})$

$\displaystyle {x}_{33}=({a}_{31},{a}_{32}) \cdot ({b}_{13},{b}_{23})=({a}_{31} {b}_{13}) +({a}_{32} {b}_{23})$

### Matrices Rules of Algebra

Matrix algebra follows a set of rules for addition and multiplication. Consider the following three square matrices: A, B, and C. A' is the inverse of A, and A-1 is the transpose of A. R is a real number, and I is the identity matrix.

Now according to the laws of matrices,

A+B = B+A $\rightarrow$ Commutative Law of Addition

A+B+C = A +(B+C) = (A+B)+C $\rightarrow$Associative law of addition

ABC = A(BC) = (AB)C $\rightarrow$ Associative law of multiplication

A(B+C) = AB + AC $\rightarrow$ Distributive law of matrix algebra

R(A+B) = RA + RB

The inverse rules of matrices are given below:

AI = IA = A

AA-1 = A-1A = I

(A-1)-1 = A

(AB)-1 = B-1A-1

(ABC)-1 = C-1B-1A-1

(A’)-1 = (A-1)’

Transpose of a Matrix

Let A = [aij]m×n. Then, A’ or AT, the transpose of A, is defined as A’ = [aji]n×m .

(i) (A’)’ = A

(ii) (λA)’ = λA’

(iii) (A+B)’ = A’+B’

(iv) (A-B)’ = A’-B’

(v) (AB)’ = A’B’

(vi) For a square matrix A, if A’ = A, then Matrix A is said to be a symmetric matrix.

(vii) For a square matrix A, if A’ = -A, then Matrix A is said to be a skew symmetric matrix.

### Application of Matrices

Matrices are used in our day-to-day life in many areas. Some of the uses of matrices in daily life are given below:

• Encryption – Encryption is a popular application of matrix in everyday life. We use it to jumble data for security reasons, and we need matrices to encode and decode it. There is a key that aids in the encoding and decoding of data generated by matrices.

• Games, Especially 3D Games – Matrices are used in games, for example, in 3D space, we use it to change the object. They convert it from a three-dimensional matrix to a two-dimensional matrix as needed.

• Economics and Business – To research a company's trends, stocks, and other assets, as well as to develop business models.

• Construction – The building industry is another common application of matrices in real life. Have you ever seen a building that is straight on the outside but the architects strive to change the structure on the outside? Matrices can be used to do this. A matrix is made up of rows and columns, and the number of rows and columns in a matrix can be changed. Matrices can assist in the support of a variety of historical structures.

• Animation – It can aid in the creation of more exact and accurate animations.

### List of Formulas

 S.no Definition Formulae 1 Order of Matrix A = [aij]m × n 2 Square Matrix m$\times$n 3 Equal Matrices aij= bij for all i and j 4. Rules for transposition of Matrices (A’)’ = A(A+B)’ = A’+B’(AB)’ = B’A’(ABC) = C’B’A’

### Solved Examples

Example 1: Given $\begin{array}{l}A= \begin{bmatrix} 3 &-5 \\ -1 &7 \end{bmatrix}\end{array}$ and $\begin{array}{l}B= \begin{bmatrix} 1 &4 \\ 8 &3 \end{bmatrix}\end{array}$ find A + B.

Solution:

Given, $\begin{array}{l}A= \begin{bmatrix} 3 &-5 \\ -1 &7 \end{bmatrix}\end{array}$

and $\begin{array}{l}B= \begin{bmatrix} 1 &4 \\ 8 &3 \end{bmatrix}\end{array}$

Addition of A and B is: $\begin{array}{l}A+B= \begin{bmatrix} 3 &-5 \\ -1 &7 \end{bmatrix}+ \begin{bmatrix} 1 &4 \\ 8 &3 \end{bmatrix}\\ =\begin{bmatrix} 3+1 &-5+4 \\ -1+8 &7+3 \end{bmatrix}\\=\begin{bmatrix} 4 &-1 \\ 7 &10 \end{bmatrix}\end{array}$

Example 2: If $\begin{array}{l}P= \begin{bmatrix} 5 &4 \\ 2 &9 \end{bmatrix}\end{array}$ and $\begin{array}{l}Q= \begin{bmatrix} 1 &5 \\ 0&2 \end{bmatrix}\end{array}$, then find P-Q.

Solution:

Given, $\begin{array}{l}P= \begin{bmatrix} 5 &4 \\ 2 &9 \end{bmatrix}\end{array}$ and

$\begin{array}{l}Q= \begin{bmatrix} 1 &5 \\ 0&2 \end{bmatrix}\end{array}$

Subtraction of the matrices P and Q is: $\begin{array}{l}A-B= \begin{bmatrix} 5 &4 \\ 2 &9 \end{bmatrix}- \begin{bmatrix} 1 &5 \\ 0 &2 \end{bmatrix}\\ =\begin{bmatrix} 5-1 &4-5 \\ 2-0 &9-2 \end{bmatrix}\\=\begin{bmatrix} 4 &-1 \\ 2 &7 \end{bmatrix}\end{array}$

### Solved Problem of Previous Year Question Paper

Question 1: $\begin{array}{l}\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \\ \end{matrix} \right];\,\,I=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]\end{array}$

$\begin{array}{l}A^{-1}=\frac{1}{6}(A^{2}+cA+dI)\end{array}$ where c, d ∈ R, then pair of values (c, d) are __________.

Solution:

Given

$\begin{array}{l}A=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \\ \end{matrix} \right]\end{array}$

$\begin{array}{l}A^{-1}=\frac{1}{6}\left[ \begin{matrix} 6 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 2 & 1 \\ \end{matrix} \right]\end{array}$

$\begin{array}{l}A^2=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \\ \end{matrix} \right]\,\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & -1 & 5 \\ 0 & -10 & 14 \\ \end{matrix} \right]\end{array}$

$\begin{array}{l} cA=\left[ \begin{matrix} c & 0 & 0 \\ 0 & c & c \\ 0 & -2c & 4c \\ \end{matrix} \right]\end{array}$

$\begin{array}{l} dI=\left[ \begin{matrix} d & 0 & 0 \\ 0 & d & 0 \\ 0 & 0 & d \\ \end{matrix} \right]\end{array}$

Therefore, by

$\begin{array}{l}A^{-1}=\frac{1}{6}(A^2+cA+dI)\end{array}$

⇒ 6 = 1 + c + d, (by equality of matrices)

So, (-6, 11) satisfy the relation.

Question 2: If $\begin{array}{l} P=\left[ \begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \end{matrix} \right],\,A=\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right]\end{array}$ and Q = PAPT, then P (Q2005)PT equal to ________.

Solution:

If Q = PAPT, PTQ = APT,

(as PPT = I) PT Q2005 P = A PT Q2004 P

= A2 PT Q2003 P

= A3 PT Q2002 P

= A2004 PT (QP)

= A2004 PT (PA) (Q = PAPT ⇒ QP = PA)

= A2005

A2005 = $\begin{array}{l}\left[ \begin{matrix} 1 & 2005 \\ 0 & 1 \\ \end{matrix} \right]\end{array}$

Question 3: If $\begin{array}{l}A=\left[ \begin{matrix} \alpha & 0 \\ 1 & 1 \\ \end{matrix} \right] and \ B=\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \\ \end{matrix} \right]\end{array}$

then value of α for which A2 = B is

A. 1

B. -1

C. 4

D. No real values

Solution:

$\begin{array}{l}A^2=\left[ \,\begin{matrix} \alpha & 0 \\ 1 & 1 \\ \end{matrix}\, \right]\,\left[ \,\begin{matrix} \alpha & 0 \\ 1 & 1 \\ \end{matrix}\, \right]=\left[ \,\begin{matrix} {{\alpha }^{2}} & 0 \\ \alpha +1 & 1 \\ \end{matrix}\, \right]\end{array}$

Clearly, no real value of a.

### Practice Questions

Q1. Let $\mathrm{M}$ and $\mathrm{N}$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $\mathrm{MN}=\mathrm{NM}$. If $\mathrm{P}^{\mathrm{T}}$ denotes the transpose of $\mathrm{P},$ then $\mathrm{M}^{2} \mathrm{N}^{2}\left(\mathrm{M}^{\mathrm{T}} \mathrm{N}\right)^{-1}\left(\mathrm{MN}^{-1}\right)^{\mathrm{T}}$ is equal to –

(A). $\mathbf{M}^{2}$

(B). $-N^{2}$

(C). $-M^{2}$

(D). MN

Q2. Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\left[\begin{array}{lll}{1} & {a} & {b} \\ {\omega} & {1} & {c} \\ {\omega^{2}} & {\omega} & {1}\end{array}\right],$ where each of a,b and $c$ is either $\omega$ or $\omega^{2} .$ Then the number of distinct matrices in the set $\mathrm{S}$ is-

(A). 2

(B). 6

(C). 4

(D). 8

### Conclusion

Matrices can be used to write and work with many linear equations, or a system of linear equations, in a compact manner. When it comes to linear transformations, often known as linear maps, matrices and matrix multiplication show their real features. We went through matrix definitions, types, and algebraic operations in this article. In addition, for a better understanding of the chapter, we went over some of the solved examples and previous year's questions.

See More

## JEE Advanced Important Dates

View All Dates
JEE Advanced 2022 exam date and revised schedule have been announced by the NTA. JEE Advanced 2022 will now be conducted on 28-August-2022, and the exam registration closes on 11-August-2022. You can check the complete schedule on our site. Furthermore, you can check JEE Advanced 2022 dates for application, admit card, exam, answer key, result, counselling, etc along with other relevant information.
See More
August
View All Dates

## JEE Advanced Information

Application Form
Eligibility Criteria
Reservation Policy
IIT Bombay has announced the JEE Advanced 2022 application form release date on the official website https://jeeadv.ac.in/. JEE Advanced 2022 Application Form is available on the official website for online registration. Besides JEE Advanced 2022 application form release date, learn about the application process, steps to fill the form, how to submit, exam date sheet etc online. Check our website for more details.
Book your Free Demo session
Get a flavour of LIVE classes here at Vedantu
Vedantu Improvement Promise
We promise improvement in marks or get your fees back. T&C Apply*

## JEE Advanced Syllabus

View JEE Advanced Syllabus in Detail
It is crucial for the the engineering aspirants to know and download the JEE Advanced 2022 syllabus PDF for Maths, Physics and Chemistry. Check JEE Advanced 2022 syllabus here along with the best books and strategies to prepare for the entrance exam. Download the JEE Advanced 2022 syllabus consolidated as per the latest NTA guidelines from Vedantu for free.
See More
Paper 1
Paper 2
View JEE Advanced Syllabus in Detail

## JEE Advanced 2022 Study Material

View all study material for JEE Advanced
JEE Advanced 2022 Study Materials: Strengthen your fundamentals with exhaustive JEE Advanced Study Materials. It covers the entire JEE Advanced syllabus, DPP, PYP with ample objective and subjective solved problems. Free download of JEE Advanced study material for Physics, Chemistry and Maths are available on our website so that students can gear up their preparation for JEE Advanced exam 2022 with Vedantu right on time.
See More
All
Chemistry
Maths
Physics
General Topics in Chemistry
Gaseous and liquid states
See All

## JEE Advanced Question Papers

see all
See More
PYQP
Sample Paper
2021
All Years
Paper 1
24th July 2021 - physics
English  •   Shift 1
24th July 2021 - chemistry
English  •   Shift 1
30th July 2021 - mathematics
English  •   Shift 1
Paper 2
30th July 2021 - physics
English  •   Shift 2
04th August 2021 - chemistry
English  •   Shift 2
04th August 2021 - mathematics
English  •   Shift 2
Paper 1
mathematics
English  •   Shift NA
mathematics
English  •   Shift NA
mathematics
English  •   Shift NA
mathematics
English  •   Shift NA
mathematics
English  •   Shift NA
Paper 2
mathematics
English  •   Shift NA
mathematics
English  •   Shift NA
mathematics
English  •   Shift NA
mathematics
English  •   Shift NA
mathematics
English  •   Shift NA

View all JEE Advanced Important Books
In order to prepare for JEE Advanced 2022, candidates should know the list of important books i.e. RD Sharma Solutions, NCERT Solutions, HC Verma books and RS Aggarwal Solutions. They will find the high quality readymade solutions of these books on Vedantu. These books will help them in order to prepare well for the JEE Advanced 2022 exam so that they can grab the top rank in the all India entrance exam.
See More
Maths
NCERT Book for Class 12 Maths
Physics
NCERT Book for Class 12 Physics
See All

## JEE Advanced 2022 Cut-Off

JEE Advanced Cut Off
IIT Bombay is responsible for the release of the JEE Advanced 2022 cut off score. The qualifying percentile score might remain the same for different categories. According to the latest trends, the expected cut off mark for JEE Advanced 2022 is 50% for general category candidates, 45% for physically challenged candidates, and 40% for candidates from reserved categories. For the general category, JEE Advanced qualifying marks for 2021 ranged from 17.50%, while for OBC/SC/ST categories, they ranged from 15.75% for OBC, 8.75% for SC and 8.75% for ST category.
See More

## Master Teachers

From IITs & other top-tier colleges with 5+ years of experience
You can count on our specially-trained teachers to bring out the best in every student.
They have taught over 4.5 crore hours to 10 lakh students in 1000+ cities in 57 countries
11+ years exp

### Shreyas

Physics master teacher

4+ years exp

### Nidhi Sharma

Chemistry master teacher

2+ years exp

### Luv Mehan

Chemistry Master Teacher

## JEE Advanced 2022 Cut-Off

JEE Advanced Cut Off
IIT Bombay is responsible for the release of the JEE Advanced 2022 cut off score. The qualifying percentile score might remain the same for different categories. According to the latest trends, the expected cut off mark for JEE Advanced 2022 is 50% for general category candidates, 45% for physically challenged candidates, and 40% for candidates from reserved categories. For the general category, JEE Advanced qualifying marks for 2021 ranged from 17.50%, while for OBC/SC/ST categories, they ranged from 15.75% for OBC, 8.75% for SC and 8.75% for ST category.
See More

## JEE Advanced 2022 Results

The JEE Advanced 2022 result will be published by IIT Bombay on https://jeeadv.ac.in/ in the form of a scorecard. The scorecard will include the roll number, application number, candidate's personal details, and the percentile, marks, and rank of the candidate. Only those candidates who achieve the JEE Advanced cut-off will be considered qualified for the exam.
See More
Rank List
Counselling
Cutoff
JEE Advanced 2022 State Wise Rank List - The respective state authorities started releasing the JEE Advanced 2022 state merit lists for admission to Engineering course. Check JEE Advanced 2022 state-wise rank list for the year 2022 here.

## JEE Advanced Top Colleges

Want to know which Engineering colleges in India accept the JEE Advanced 2022 scores for admission to Engineering? Find the list of Engineering colleges accepting JEE Advanced scores in India, compiled by Vedantu. There are 1622 Colleges that are accepting JEE Advanced. Also find more details on Fees, Ranking, Admission, and Placement.
See More