NCERT Class 12 Maths Chapter 4 Notes, Determinants Class 12 Chapter 4 Notes

Have you ever wondered how characters and objects move and stay stable in 3D video games? Or, you have to go some place emergency and you search in google map for the shortest route. Have you ever thought about which mechanism maps are determining the shortest route? In class 12 maths chapter 12 notes, we learn about various properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle, adjoint and inverse of a square matrix, and some other important concepts.

Determinants are useful in many fields like engineering, navigation, cryptography, and computer graphics. These NCERT Class 12 Maths Chapter 4 Notes will give a better conceptual clarity and deeper understanding of the topic. These NCERT notes are well structured and prepared by experienced Careers360 experts.

Determinant

To every square matrix $\mathrm{A}=\left[a_{ij}\right]$ of order $n$, we can associate a number (real or complex) called the determinant of the matrix A, written as $\det \mathrm{A}$, where $a_{ij}$ is the $(i,j)th$ element of A.
If $\mathrm{A}=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$, then determinant of A , denoted by $|\mathrm{A}|($ or $\det \mathrm{A})$, is given by

$|\mathrm{A}|=\left|\begin{array}{ll}a& b\\ c& d\end{array}\right|=ad-bc$

Let two equations $a_{1}x+b_1=0$ and $a_{2}x+b_2=0$ are satisfied by the same value of x.

$\Rightarrow x=\frac{-b{}_1}{a_1}=\frac{-b{}_2}{a_2}$

$\Rightarrow a_1{b}_2-a_2{b}_1=0$

The expression ${\mathrm{a}}_1{\mathrm{b}}_2-{\mathrm{a}}_2{\mathrm{b}}_1$ is called a determinant of the second-order and is denoted by :

$D=\left|\begin{array}{ll}{a}_1& b_1\\ a_2& b_2\end{array}\right|$

Let's consider the system of equations

$a_{1}x+b_{1}y+c_1=0$

$a_{2}x+b_{2}y+c_2=0$

$a_{3}x+b_{3}y+c_3=0$

If these equations are satisfied by the same x and y values, then eliminate x and y.

$a_1(b_2{c}_3-b_3{c}_2)+b_1(c_2{a}_3-c_3{a}_2)+c_1(a_2{b}_3-a_3{b}_2)=0$

The expression $a_1(b_2{c}_3-b_3{c}_2)+b_1(c_2{a}_3-c_3{a}_2)+c_1(a_2{b}_3-a_3{b}_2)$ is called a determinant of the third-order and is denoted by :

$D=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

Note- Determinants consist of an equal number of rows and columns.

Value of a Determinant:

Determinant of a matrix of order one:-

Let $\mathrm{A}=[a]$ be the matrix of order 1, then the determinant of A is defined to be equal to $a$.

Determinant of a matrix of order two:-

$\mathrm{A}=\left[\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$

$\text{ det }(\mathrm{A})=|\mathrm{Al}=\mathrm{\Delta }=\left|\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$

$\text{ det }(\mathrm{A})=a_{11}{a}_{22}-a_{21}{a}_{12}$

Determinant of a matrix of order three:-

$A=\left[\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$

$det(A)=D=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

$D=a_1\left|\begin{array}{ll}{b}_2& c_2\\ b_3& c_3\end{array}\right|-b_1\left|\begin{array}{ll}{a}_2& c_2\\ a_3& c_3\end{array}\right|+c_1\left|\begin{array}{ll}{a}_2& b_2\\ a_3& b_3\end{array}\right|$

$D=a_1(b_2{c}_3-b_3{c}_2)-b_1(a_2{c}_3-a_3{c}_2)+c_1(a_2{b}_3-a_3{b}_2)$

Properties of Determinants:

• The value of a determinant doesn't change if the rows & columns are interchanged
  $D=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{lll}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$
• If any two rows of a determinant are interchanged, the sign of the value of the determinant is changed
  Let $\mathrm{D}=\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$
  and ${\mathrm{D}}^{\prime }=\left|\begin{array}{lll}{\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$
  then ${\mathrm{D}}^{\prime }=-\mathrm{D}$
• If any two columns of a determinant are interchanged, the sign of the value of the determinant is changed
  Let $\mathrm{D}=\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$
  and${\mathrm{D}}^{\prime }=\left|\begin{array}{lll}{\mathrm{b}}_1& {\mathrm{a}}_1& {\mathrm{c}}_1\\ {\mathrm{b}}_2& {\mathrm{a}}_2& {\mathrm{c}}_2\\ {\mathrm{b}}_3& {\mathrm{a}}_3& {\mathrm{c}}_3\end{array}\right|$
  then ${\mathrm{D}}^{\prime }=-\mathrm{D}$
• If all the elements of a row are zero, then the value of the determinant is zero.
  $\mathrm{D}=\left|\begin{array}{lll}0& 0& 0\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|=0$
• If all the elements of a column are zero, then the value of the determinant is zero.
  $\mathrm{D}=\left|\begin{array}{lll}0& {\mathrm{b}}_1& {\mathrm{c}}_1\\ 0& {\mathrm{b}}_2& {\mathrm{c}}_2\\ 0& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|=0$
• If all the elements of any row are multiplied by the same number, then the determinant is multiplied by that number.
  Let $\mathrm{D}=\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$
  and ${\mathrm{D}}^{\prime }=\left|\begin{array}{lll}k{\mathrm{a}}_1& k{\mathrm{b}}_1& k{\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$
  then ${\mathrm{D}}^{\prime }=k\mathrm{D}$
• If all the elements of any column are multiplied by the same number, then the determinant is multiplied by that number.
  Let $\mathrm{D}=\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$
  and ${\mathrm{D}}^{\prime }=\left|\begin{array}{lll}k{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ k{\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ k{\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$
  then ${\mathrm{D}}^{\prime }=k\mathrm{D}$
• If all the elements of a row are proportional to the element of any other row, then the determinant's value is zero.
  $\mathrm{D}=\left|\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ k{\mathrm{a}}_1& k{\mathrm{b}}_1& k{\mathrm{c}}_1\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|=0$
• If all the elements of a column are proportional to the element of any other column, then the determinant's value is zero.
  $\mathrm{D}=\left|\begin{array}{lll}k{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ k{\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ k{\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$
• If some or all elements of a row or column of a determinant are expressed as the sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants.
  $D=\left|\begin{array}{ccc}{a}_1+p& a_2+q& a_3+r\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|=\left|\begin{array}{lll}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|+\left|\begin{array}{lll}p& q& r\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$
• The value of the determinant remains the same if we apply the operation $R_i\to R_i+k{R}_{j}or{C}_i\to C_i+k{C}_j$

Area of a Triangle:

The area of a triangle whose vertices are $(x_1,y_1),(x_2,y_2)\text{ and }(x_3,y_3)$ is given by:

$\mathrm{\Delta }=\frac{1}{2}\left|\begin{array}{lll}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$

Minors of Determinants:

The minor of a given element of the determinant is the determinant obtained by deleting the row & the column in which the given element stands.

Let $D=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

The minor of element a1 $M_{11}=\left|\begin{array}{ll}{b}_2& c_2\\ b_3& c_3\end{array}\right|$

The minor of element b1 =$M_{12}=\left|\begin{array}{ll}{a}_2& c_2\\ a_3& c_3\end{array}\right|$

The minor of element c1 = $M_{13}=\left|\begin{array}{ll}{a}_2& b_2\\ a_3& b_3\end{array}\right|$

Here Mij represents the minor of the element belonging to the ith row and jth column.

Cofactors of Determinants:

The cofactor of element ith row and jth column is given by: $C_{ij}=(-1{)}^{i+j}{M}_{ij}$ where Mij represents the minor of the element belonging to ith row and jth column.

The cofactor of element a1 $C_{11}=\left|\begin{array}{ll}{b}_2& c_2\\ b_3& c_3\end{array}\right|$

The cofactor of element b1 =$C_{12}=(-1)\left|\begin{array}{ll}{a}_2& c_2\\ a_3& c_3\end{array}\right|$

The cofactor of element c1 = $C_{13}=\left|\begin{array}{ll}{a}_2& b_2\\ a_3& b_3\end{array}\right|$

Expansion of Determinants in terms of Elements of any Row or Column:

Let $D=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$

The sum of the product of elements of any row (column) with their corresponding cofactors is always equal to the value of the determinant.

So the determinant D can be in 6 forms

$D=a_1{C}_{11}+b_1{C}_{12}+c_1{C}_{13}$

$D=a_2{C}_{21}+b_2{C}_{22}+c_2{C}_{23}$

$D=a_3{C}_{31}+b_3{C}_{32}+c_3{C}_{33}$

$D=a_1{C}_{11}+a_2{C}_{21}+a_3{C}_{31}$

$D=b_1{C}_{12}+b_2{C}_{22}+b_3{C}_{32}$$D=c_1{C}_{13}+c_2{C}_{23}+c_3{C}_{33}$

Adjoint of a Matrix:

Let a given matrix $A=\left[\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$ then adjoin of matrix A is defined as

$\text{ adj(A) = Transpose of }\left[\begin{array}{lll}{\mathrm{C}}_{11}& {\mathrm{C}}_{12}& {\mathrm{C}}_{13}\\ {\mathrm{C}}_{21}& {\mathrm{C}}_{22}& {\mathrm{C}}_{23}\\ {\mathrm{C}}_{31}& {\mathrm{C}}_{32}& {\mathrm{C}}_{33}\end{array}\right]=\left[\begin{array}{lll}{\mathrm{C}}_{11}& {\mathrm{C}}_{21}& {\mathrm{C}}_{31}\\ {\mathrm{C}}_{12}& {\mathrm{C}}_{22}& {\mathrm{C}}_{32}\\ {\mathrm{C}}_{13}& {\mathrm{C}}_{23}& {\mathrm{C}}_{33}\end{array}\right]$

where - Cij is the cofactor of the element aij

Note: For any given matrix of order n

$\mathrm{A}(adj\mathrm{A})=(adj\mathrm{A})\mathrm{A}=|\mathrm{A}|\mathrm{I}$

where I - Identity matrix of order n

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 $|A|\ne 0$

The inverse of a Matrix:

A square matrix A is invertible if and only if A is a nonsingular matrix, or in other words, a square matrix A is invertible if and only if $|A|\ne 0$.

$\mathrm{A}(adj\mathrm{A})=(adj\mathrm{A})\mathrm{A}=|\mathrm{A}|\mathrm{I}$

${\mathrm{A}}^{-1}=\frac{1}{|\mathrm{A}|}adj\mathrm{A}$

Solution of a System of Linear Equations using Inverse of a Matrix:

Let us give the system of linear equations

$\begin{array}{rl}& a_{1}x+b_{1}y+c_{1}z=d_1\\ & a_{2}x+b_{2}y+c_{2}z=d_2\\ & a_{3}x+b_{3}y+c_{3}z=d_3\end{array}$

Let $\mathrm{A}=\left[\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$

$\mathrm{X}=\left[\begin{array}{l}x\\ y\\ z\end{array}\right]$

$\mathrm{B}=\left[\begin{array}{l}{d}_1\\ d_2\\ d_3\end{array}\right]$

Then the system of the linear equation can be written as

AX = B

$\left[\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}{d}_1\\ d_2\\ d_3\end{array}\right]$

- If A is a non-singular matrix

$\mathrm{X}={\mathrm{A}}^{-1}\mathrm{B}$

- If A is a singular matrix

• If $adj(A).B\ne 0$ then the linear equation does not have solutions, and is an inconsistent linear equation
• If $adj(A).B=0$ then linear equations are either consistent or inconsistent and the system has either infinitely many solutions or no solution.

Note-1 If A and B are square matrices of the same order, then the determinant of the product of matrices is equal to the product of their respective determinants, i.e.

$|AB|=|A|.|B|$

Note-2 If A is a square matrix of order n, then

$|adj(A)|=|A{|}^{n-1}$

System of linear equations

(i) Consider the equations:

$\begin{array}{rl}& a_{1}x+b_{1}y+c_{1}z=d_1\\ & a_{2}x+b_{2}y+c_{2}z=d_2\\ & a_{3}x+b_{3}y+c_{3}z=d_3,\end{array}$

In matrix form, these equations can be written as $\mathrm{AX}=\mathrm{B}$, where

$\mathrm{A}=\left[\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right],\mathrm{X}=\left[\begin{array}{l}x\\ y\\ z\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}{d}_1\\ d_2\\ d_3\end{array}\right]$

(ii) Unique solution of equation $\mathrm{AX}=\mathrm{B}$ is given by $\mathrm{X}={\mathrm{A}}^{-1}\mathrm{B}$, where $|\mathrm{A}|\ne 0$.

(iii) A system of equations is consistent or inconsistent according as its solution exists or not.
(iv) For a square matrix A in matrix equation $\mathrm{AX}=\mathrm{B}$
(a) If $|\mathrm{A}|\ne 0$, then there exists unique solution.
(b) If $|\mathrm{A}|=0$ and $(\mathrm{adj}\mathrm{A})\mathrm{B}\ne 0$, then there exists no solution.
(c) If $|\mathrm{A}|=0$ and $(\mathrm{adj}\mathrm{A})\mathrm{B}=0$, then system may or may not be consistent.

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Important points to note

• NCERT problems are very important for performing well in exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT Solutions for Class 12 Maths Chapter 4 Determinants

• Students are advised to go through the Class 12 Maths Chapter 4 Notes before solving the questions.

• To boost your exam preparation as well as for quick revision, these NCERT notes are very useful.