JEE Advanced 2024 Matrices and Determinants Notes

Matrices:

Two matrices are said to be equal if they have the same order, and each element of one is equal to the corresponding element of the other.
A matrix is a rectangular array of $m \times n$ numbers arranged in $m$ rows and $n$ columns.

${\left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}...............}&{{a_{1n}}} \\{{a_{21}}}&{{a_{22}}..............}&{{a_{2n}}} \\{{a_{m1}}}&{{a_{m2}}.............}&{{a_{mn}}} \end{array}} \right)_{m \times n}}$

Or 

$A = {\left[ {{a_{ij}}} \right]_{m \times n}}$ where, $i = 1,2,......,m;$ $j = 1,2,......,n.$

Row Matrix: A matrix which has one row is called row matrix. $A = {\left[ {{a_{ij}}} \right]_{1 \times n}}$

Column Matrix: A matrix which has one column is called column matrix. $A = {\left[ {{a_{ij}}} \right]_{m \times 1}}$

Square Matrix: A matrix in which number of rows are equal to number of columns, is called a square matrix. $A = {\left[ {{a_{ij}}} \right]_{m \times n}}$.

Diagonal Matrix: A square matrix is called diagonal matrix if all the elements, except the diagonal elements are zero.

Scalar Matrix: A square matrix is called scalar matrix it all the elements, except diagonal elements are zero and diagonal elements are equal.

$A = {\left[ {{a_{ij}}} \right]_{m \times n}}$, Where${a_{ij}} = 0,i \ne j$,

${a_{ij}} = \alpha ,i = j$.

Identity or Unit Matrix: A square matrix in which all the non-diagonal elements are zero and diagonal elements are unity is called identity or unit matrix.

Null Matrix: A matrix in which all elements are zero.

Equal Matrix: Two matrices are said to be equal if they have the same order and all their corresponding elements are equal.

Transpose of the Matrix:

If $A$ is the given matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of a matrix.

Properties of Transpose: If $A$ and $B$ are matrices such that their sum and product are defined, then 

1. ${({A^T})^T} = A$

2. ${\left( {A + B} \right)^T} = {A^T} + {B^T}$

3. $\left( {K{A^T}} \right) = K{A^T}$ Where, K is a scalar.

4. ${\left( {AB} \right)^T} = {B^T}{A^T}$

5. ${\left( {ABC} \right)^T} = {C^T}{B^T}{A^T}$

Orthogonal Matrix: A square matrix pf order $n$ is said to be orthogonal, if $AA’= {I_n} = A’A$

Properties of the Orthogonal Matrix:

1. If $A$ is an orthogonal matrix, then $A’$  is also an orthogonal matrix.

2. For any two orthogonal matrices, $A$ and $B$, $AB$, and $BA$ is also an orthogonal matrix.

3. If $A$ is an orthogonal matrix, ${A^{ - 1}}$ is also orthogonal matrix.

Idempotent Matrix: A square matrix $A$ is said to be idempotent, if ${A^2} = A$.

Properties of Idempotent Matrix: If $A$ and $B$ are two idempotent matrices, then

$AB$ is idempotent, if $AB$=$BA$.

$A + B$ is an idempotent matrix, if $AB = BA = 0$

$AB = A$ and $BA = B$, then ${A^2} = A,{B^2} = B$

If $A$ is an idempotent matrix and $A + B = I$, then $B$ is an idempotent and $AB = BA = 0$.

Diagonal $\left( {1,1,1,........,1} \right)$ is an idempotent matrix.

If ${I_1},{I_2}$ and ${I_3}$ are direction cosines, then

$\left( {\begin{array}{*{20}{c}}{l_1^2}&{{l_1}{l_2}}&{{l_1}{l_3}} \\{{l_1}{l_2}}&{l_2^2}&{{l_2}{l_3}} \\{{l_3}{l_1}}&{{l_3}{l_2}}&{l_3^2}\end{array}} \right)$

Is an idempotent as $|\Delta {|^2} = 1$

A square matrix $A$ is said to be involuntary, if ${A^2} = I$.

Nilpotent Matrix: A square matrix $A$ is said to be a nilpotent matrix, if there exists a positive integer m such that ${A^2} = 0$. If $m$ is the least positive integer such that ${A^m} = 0$, then m is called the index of the nilpotent matrix $A$.

Unitary Matrix: A square matrix $A$ is said to be unitary if $A’A = I$.

Hermitian Matrix: A square matrix $A$ is said to be hermitian matrix, if $A = {A^.}$ or $ = {a_{ij}}$, for $= {a_{ij}}$ only.

Some Chief Properties of Matrices:

1. Only matrices of the same order can be added or subtracted.

2. The addition of matrices is commutative as well as associative.

3. Cancellation laws hold well in case of addition.

4. The equation $A + X = 0$ has a unique solution in the set of all $m \times n$ matrices.

5. All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by a scalar.

Symmetric Matrix:

For example $A = \left( {\begin{array}{*{20}{c}}{\sqrt 3 }&2&3 \\2&{ - 1.5}&{ - 1} \\3&{ - 1}&1\end{array}} \right)$ is a symmetric matrix as $A' = A$

Skew Symmetric Matrix:

A square matrix is said to be skew symmetric if ${A^T} = A$.

If $A = {\left[ {{a_{ij}}} \right]_{m \times n}}$, then ${a_{ij}} =  - {a_{ij}}$ for all $i,j$.

For example, the matrix $B = \left( {\begin{array}{*{20}{c}}0&e&f \\{ - e}&0&g \\{ - f}&{ - g}&0\end{array}} \right)$ is a skew matrix as $B' = B$.

Multiplication of a Matrix by a Scalar:

If $A = {[{a_{ij}}]_{m \times n}}$ is a matrix and $k$ is a scalar. Then $kA$ is another matrix which is obtained by multiplying each element of $A$ by the scalar $k$

For example $A = \left( {\begin{array}{*{20}{c}}1&3&5 \\5&9&5 \\8&2&4\end{array}} \right)$ then $2A = \left( {\begin{array}{*{20}{c}}2&6&{10} \\{10}&{18}&{10} \\{16}&4&8\end{array}} \right)$

If $A = {\left[ {{a_{ij}}} \right]_{m \times n}}$, then ${a_{ij}} =  - {a_{ij}}$ for all$i,j$.

Product of Matrices:

If A & B are two matrices, product $AB$ is defined. If the number of columns of $A =$ number of rows of $B$.

Let $A = {\left[ {{a_{ij}}} \right]_{m \times n}}$, and $B = {\left[ {{b_{jk}}} \right]_{n \times p}}$ then $AB$ is the matrix $C$ of the order mxp. to get ${\left( {i,k} \right)^{th}}$ element ${c_{ik}}$ of the matrix $C$ we take ${i^{th}}$ row of $A$ and ${k^{th}}$ column of $B$ multiply them element wise and take the sum of all these products.

Example If $\mathrm{A}=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$, then

(A) Adj $A$ is a zero matrix

(B) Adj $A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right]$

(C) $A^{-1}=A$

(D) $A^{2}=1$

Ans: (B, C)

(A) is false, and since most elements in $A$ are 0, the adjoint can be easily found.

We note that adj $A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right]$

$\Rightarrow$ Choice (B) is correct.

Now $A^{-1}=\dfrac{\operatorname{adj} A}{|A|}=\operatorname{adj} A \quad(-|A|=1)$

$\Rightarrow$ Choice (C) is also true.

If $\mathrm{A}^{2}=1$ then $\mathrm{A}=\mathrm{A}^{-1}=\mathrm{Adj} \mathrm{A}$ which is not true.

$\Rightarrow$ Choice (D) is not true.

Determinants:

To every square matrix we can assign a number called its determinant.

$A = \left[ {{a_{11}}} \right]$, det.$A = |A| = {a_{11}}$.

If $A = \left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}} \\{{a_{21}}}&{{a_{22}}}\end{array}} \right)$, 

$|A| = {a_{11}}{a_{22}} - {a_{21}}{a_{12}}$.

Properties:

• The value of a determinant remains unchanged if its rows and columns are interchanged.

• If any two rows/columns, then the sign of the determinant changes.

• If two rows/columns of a determinant are the identical value of the determinant is zero.

• If all the elements of the rows/columns of a determinant are multiplied by a constant $k$, then its value gets multiplied by $k$.

• If elements of any one column are expressed as the sum of two elements each, then determinants can be written as the sum of two determinants.

• If $A\& B$ are square matrices of same order, then $|AB| = |A||B|$.

Minors and Cofactors:

Definition:

Minor of an element ${a_{ij}}$ of a determinant id the determinant obtained by deleting its ith row and jth column in which element ${a_{ij}}$ lies. Minor of an element ${a_{ij}}$ is denoted by ${M_{ij}}$.

Cofactor of an element ${a_{ij}}$, denoted by ${A_{ij}}$ is defined by ${A_{ij}} = {\left( { - 1} \right)^{i + j}}{M_{ij}}$, Where ${M_{ij}}$ is minor of ${a_{ij}}$.

Singular Matrix: A square matrix $A$ of order $n$ is said to be singular, if $|A| = 0$.

Non-Singular Matrix: A square matrix $A$ of order $n$ is said to be non-singular, if $|A| \ne 0$.

Adjoint of the Matrix:

If $A = \left[ {{a_{ij}}} \right]$ be a n-square matrix then transpose of cofactors of element of matrix $A$, is called the adjoint of $A$.

Adj. $A = {\left[ {{A_{ij}}} \right]^T}$.

Invertible Matrix:

A matrix $A$ is said to be invertible if another matrix $B$ exists, such as $AB = BA = I$.

The Inverse of a Matrix:

The inverse of a square matrix $A$ exists if $A$ is non-singular.

${A^{ - 1}} = \dfrac{1}{{|A|}}Adj.A$

Properties of the Inverse of a Matrix:

1. Every invertible matrix possesses a unique inverse.

2. If $A$ and $B$ are invertible matrices of the same order, then $AB$ is invertible and ${\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}$. This is also termed as the reversal law.

3. In general, if $A,B,C,...$ are invertible matrices then ${\left( {ABC...} \right)^{ - 1}} = .....{C^{ - 1}}{B^{ - 1}}{A^{ - 1}}$.

4. If $A$ is an invertible square matrix, then ${A^T}$ is also invertible and ${\left( {{A^T}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T}$. 

Q. If $\mathrm{A}=\left(\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right)$ and $\mathrm{A}^{2}-4 \mathrm{~A}-5 \mathrm{I}=\mathrm{O}$ where $\mathrm{I}$ and $\mathrm{O}$ are the unit matrix and the null matrix of order 3 respectively. If $15 A^{-2}=2\left|\begin{array}{ccc}-3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3\end{array}\right|$ then the find the value of $\lambda$.

Ans: 3

$A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$

$A^{2}=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]=\left[\begin{array}{lll}9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9\end{array}\right]$

$4 A-\left[\begin{array}{lll}4 & 8 & 8 \\8 & 4 & 8 \\8 & 8 & 4\end{array}\right]$

$A^{2}-4 A-51=0$

$\left[\begin{array}{lll}9 & 8 & 8 \\8 & 9 & 8 \\8 & 8 & 9\end{array}\right]-\left[\begin{array}{lll}4 & 8 & 8 \\8 & 4 & 8 \\8 & 8 & 4\end{array}\right]-\left[\begin{array}{lll}5 & 0 & 0 \\0 & 5 & 0 \\0 & 0 & 5\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & 0\end{array}\right]=0$

$A^{2}-4 A-51=0$

Multiplying by $A^{-1}$ both sides,

$A^{-1}\left[A^{2}-4 A-51\right]-A^{-2} 0$

$A-41-5 A^{-1}=0$

$A-41=5 A^{-1}$

$A^{-1}=\dfrac{1}{5}(A-41)=\dfrac{1}{5}\left(\left[\begin{array}{lll}1 & 2 & 2 \\2 & 1 & 2 \\2 & 2 & 1\end{array}\right]-\left[\begin{array}{lll}4 & 0 & 0 \\0 & 4 & 0 \\0 & 0 & 4\end{array}\right]\right)$

$A^{-1}=\dfrac{1}{5}\left[\begin{array}{ccc}-3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3\end{array}\right]$

System of Linear Equations:

${a_1}x + {b_1}y + {c_1}z = {d_1}$

${a_2}x + {b_2}y + {c_2}z = {d_2}$

${a_3}x + {b_3}y + {c_3}z = {d_3}$

In matrix from the above equations can be written as

$\left( {\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}} \\{{a_2}}&{{b_2}}&{{c_2}} \\{{a_3}}&{{b_3}}&{{c_3}}\end{array}} \right)$

$\left[\begin{array}{l}x \\y \\z\end{array}\right]=\left[\begin{array}{l}d_{1} \\d_{2} \\d_{3}\end{array}\right]$

That is $A X=B$

$X=A^{-1} B$

Criteria of Consistency:

1. If $|A| \ne 0$, then the system of equations is said to be consistent & has a unique solution.

2. If $|A| = 0$ and $\left( {adjA} \right)b = 0$, Then the system of equations is consistent and has infinitely many solutions.

3. If $|A| = 0$ and $\left( {adjA} \right)b \ne 0$, then the system of equations is inconsistent and has no solution.

Examples:

If $\left( {\begin{array}{*{20}{c}}{x + 3}&{z + 4}&{2y - 7} \\{ - 6}&{a - 1}&0 \\{b - 3}&{ - 21}&0\end{array}} \right) = \left( {\begin{array}{*{20}{c}}0&6&{3y - 2} \\{ - 6}&{ - 3}&{2c + 2} \\{2b + 4}&{ - 21}&0\end{array}} \right)$, then find the value of a, b, c, x, y and z.

Solution:

It is given that, the two matrices are equal. Therefore, the corresponding elements present in matrices should be equal to each other. By comparing the corresponding elements in the matrices, we get:

$x + 3 = 0$

$\Rightarrow x =  - 3$

$z + 4 = 6$

$\Rightarrow z = 6 - 4$

$z = 2$

Since, $2y - 7 = 3y - 2$

Therefore, $3y - 2y =  - 7 + 2$

$\Rightarrow y =  - 5$

Also, $a - 1 =  - 3$

Therefore, $a =  - 2$

Similarly, $2c + 2 = 0$

$2c =  - 2$

$c =  - 1$

Now on comparing, $b - 3 = 2b + 4$

We get,

$2b - b =  - 3 - 4$

$b =  - 7$

Therefore, the values of the variables are:

a =  - 2

b =  - 7

c =  - 1

x =  - 3

y =  - 5

z = 2

Example: If $A=\left[\begin{array}{c}1 \\ 3 \\ -6\end{array}\right]$ and $B=\left[\begin{array}{ll}-2 & 4\end{array}\right]$. Then verify that $(A B)^{T}=B^{T} A^{T} .$

Solution:

Given,

$A=\left[\begin{array}{c}1 \\3 \\-6\end{array}\right]$

And

$B=\left[\begin{array}{lll}-2 & 4 & 5\end{array}\right]$

$A B=\left[\begin{array}{c}1 \\ 3 \\ -6\end{array}\right]$

$B=\left[\begin{array}{lll}-2 & 4 & 5\end{array}\right]$

$= \left( {\begin{array}{*{20}{c}}{ - 2}&4&5 \\{ - 6}&{12}&{15} \\{12}&{ - 24}&{ - 30}\end{array}} \right)$

Now, we need to calculate the transpose of $AB$

${\left( {AB} \right)^T} = \left( {\begin{array}{*{20}{c}}{ - 2}&4&5 \\{ - 6}&{12}&{15} \\{12}&{ - 24}&{ - 30}\end{array}} \right)$

${A^T} = [1$ $3$ $ - 6]$

And

$B^{T}=\begin{bmatrix}-2\\4\\5\end{bmatrix}$

$B^{T}A^{T}=\begin{bmatrix}-2\\4\\5\end{bmatrix}$

$=\begin{bmatrix}1~~~~~3~-6\end{bmatrix}$

$= \left( {\begin{array}{*{20}{c}}{ - 2}&4&5 \\{ - 6}&{12}&{15} \\{12}&{ - 24}&{ - 30}\end{array}} \right)$

Therefore, ${\left( {AB} \right)^T} = {B^T}{A^T}$.

Hence verified.

Example: If $A = \left( {\begin{array}{*{20}{c}}2&{ - 1}&1 \\{ - 2}&1&{ - 3} \\0&1&4\end{array}} \right)$. Find ${A^{ - 1}}$ using elementary operation.

Solution:

$A = IA$

$\left( {\begin{array}{*{20}{c}}2&{ - 1}&1 \\{ - 2}&1&{ - 3} \\0&1&4\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&0&0 \\0&1&0 \\0&0&1\end{array}} \right)A$

${R_1} \to {R_1} + {R_3}$

$\left( {\begin{array}{*{20}{c}}2&{ - 1}&1 \\{ - 2}&1&{ - 3} \\0&1&4\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&0&0 \\0&1&0 \\0&0&1\end{array}} \right)A$

${R_2} \to {R_2} + {R_1}$

$\left( {\begin{array}{*{20}{c}}2&0&5 \\0&1&2 \\0&1&4\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&0&0 \\0&1&0 \\ 0&0&1\end{array}} \right)A$  

${R_3} \to {R_3} - {R_2}$

$\left( {\begin{array}{*{20}{c}}2&0&5 \\0&1&2 \\0&1&4\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&0&0 \\0&1&0 \\0&0&1\end{array}} \right)A$

${R_1} \to 2{R_1} - 5{R_2}$,  ${R_1} \to {R_1} - {R_3}$

$\left( {\begin{array}{*{20}{c}}4&0&0 \\0&1&0 \\0&0&2\end{array}} \right) = \left( {\begin{array}{*{20}{c}}7&5&2 \\2&2&1 \\{ - 1}&{ - 1}&0 \end{array}} \right)A$

${R_1} \to \dfrac{1}{4}{R_1},{R_3} \to \dfrac{1}{2}{R_3}$

$\left( {\begin{array}{*{20}{c}}1&0&0 \\0&1&0 \\0&0&1\end{array}} \right) = \left( {\begin{array}{=*{20}{c}}{\dfrac{7}{4}}&{\dfrac{5}{4}}&{\dfrac{1}{2}} \\2&2&1 \\{\dfrac{{ - 1}}{4}}&{\dfrac{{ -1}}{4}}&0\end{array}} \right)A$

${A^{-1}}=\left({\begin{array}{*{20}{c}}{\dfrac{7}{4}}&{\dfrac{5}{4}}&{\dfrac{1}{2}} \\2&2&1 \\{\dfrac{{ - 1}}{4}}&{\dfrac{{ - 1}}{4}}&0\end{array}} \right)$

Importance of JEE Advanced Maths Matrices and Determinants

Different objects, numbers, and alphabets are arranged in rectangular arrays to perform various algebraic operations, these operations are done by following specific rules. This chapter explains how complex algebraic operations can be done in an easier way by following these matrix and determinant formulas.

This chapter is a crucial part of linear algebra. Linear equations can also be solved using the methods explained in this chapter. When the equations are not homogenous then these methods and formulas are used to calculate the result. In this section, you will be introduced to square matrices to calculate determinants and other related sums.

All the terms related to this chapter will be explained using proper mathematical derivations and examples. You will learn how to fill the rows and columns and perform various mathematical operations related to matrices and determinants.

This chapter teaches how to solve a system of homogenous and non-homogenous algebraic equations in the form of matrices and determinants. Matrices and Determinants JEE Advanced revision notes will become very useful for you to understand all the formulas and steps used to perform such mathematical operations.

Benefits of Matrices and Determinants JEE Advanced Notes PDF

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• The easier explanation of the algebraic operations such as subtraction, addition, and multiplication will guide you to prepare this chapter faster.

• Imbibe the concepts of this chapter easily and learn how to use them in different subjects. This chapter is very important for JEE Advanced. All these formulas and concepts have applications in advanced concepts of mathematics. Hence, the addition of revision notes to the study material will help you to better prepare for the exam.

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Students can download the free PDF of revision notes from our website or mobile app and refer to them at their convenience. In these notes, our experts have explained the formulas and operations of matrices to help students develop a strong conceptual foundation for these topics. Also, students can solve the sample questions given in the revision notes and assess their knowledge of this chapter. Hence, students can refer to these revision notes on Matrices and Determinants available on Vedantu to prepare this chapter effectively.

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