Determinants Class 12th Notes - Free NCERT Class 12 Maths Chapter 4 Notes - Download PDF

NCERT Class 12 Maths Chapter 4 Determinants Notes - The determinant of the matrix is the factor by which its volume blows up. In Determinants Class 12 Note, you will learn about important topics like determinants and their properties, finding the area of the triangle, etc. Also, minor and co-factors, adjoint and the inverse of the matrix, and applications of determinants like solving the system of linear equations, etc are included in the CBSE Class 12 Maths Chapter 4 Notes. This chapter is very important for the students in CBSE board exams as well as competitive exams like JEE and NEET. Students are advised to thorough with NCERT textbook. You can take help from these NCERT Class 12 Maths Chapter 4 Notes if you are finding difficulty in understanding the concepts. Here, you can also check NCERT Book.

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Determinant:

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 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 values of x and y, then on eliminating 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 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 inter-changed
  $D=\left|\begin{array}{lll}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{lll}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$
• 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-column 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 M_ij represents the minor of the element belonging to 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 M_ij 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 - C_ij is the cofactor of the element a_ij

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 $|A|\ne 0$

Inverse of a Matrix:

A square matrix A is invertible if and only if A is 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 we have given the system of linear equation

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

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 linear equation does not have solutions and inconsistent linear equation
• If $adj(A).B=0$ then linear equations are either consistent or inconsistent and the system have 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

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

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