Determinants

What are Determinants?

The determinant of a matrix A is a number that is calculated from the matrix. For a determinant to exist, matrix $A$ must be a square matrix. Matrix is only a representation while determinant is the value of the matrix. The determinant of a matrix is denoted by $\operatorname{det} \mathrm{A}$ or $|\mathrm{A}|$.

To every square matrix $A=\left[a_{i j}\right]$ of order $n$, we can associate a number called determinant of the matrix $A$.

If $A=\left[\begin{array}{cccc}a_{11} & a_{12} & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n}\end{array}\right]$, then determinant of $A$ is written as $|A|=\left|\begin{array}{cccc}a_{11} & a_{12} & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n}\end{array}\right|$.

Minors

Let $A=\left[a_{i j}\right]_{3 \times 3}$ be a given square matrix of order $3$ . The minor of an arbitrary element $a_{i j}$ is the determinant obtained by deleting the $i^{\text {th }}$ row and $j^{\text {th }}$ column in which the element $a_{i j}$ stands. The minor of $a_{i j}$ is usually denoted by $M_{i j}$.

Cofactors

The cofactor is a signed minor. The cofactor of $a_{i j}$ is usually denoted by $A_{i j}$ and is defined as $A_{i j}=(-1)^{i+j} M_{i j}$.

For instance, consider the $3 \times 3$ matrix defined by $A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$
Then the minors and cofactors of the elements $a_{11}, a_{12}, a_{13}$ are given as follows :
(i) Minor of $a_{11}$ is $\mathrm{M}_{11} \quad=\left|\begin{array}{ll}a_{22} & a_{23} \\ a_{32} & a_{33}\end{array}\right|=a_{22} a_{33}-a_{32} a_{23}$

Cofactor of $a_{11}$ is $A_{11}=(-1)^{1+1} M_{11}=\left|\begin{array}{ll}a_{22} & a_{23} \\ a_{32} & a_{33}\end{array}\right|=a_{22} a_{33}-a_{32} a_{23}$
(ii) Minor of $a_{12}$ is $M_{12} \quad=\left|\begin{array}{ll}a_{21} & a_{23} \\ a_{31} & a_{33}\end{array}\right|=a_{21} a_{33}-a_{31} a_{23}$

Cofactor $a_{12}$ is $A_{12} \quad=(-1)^{1+2}\left|\begin{array}{ll}a_{21} & a_{23} \\ a_{31} & a_{33}\end{array}\right|=-\left(a_{21} a_{33}-a_{31} a_{23}\right)$
(iii) Minor of $a_{13}$ is $M_{13} \quad=\left|\begin{array}{ll}a_{21} & a_{22} \\ a_{31} & a_{32}\end{array}\right|=a_{21} a_{32}-a_{31} a_{22}$

Cofactor of $a_{13}$ is $A_{13}=(-1)^{1+3} M_{13}=\left|\begin{array}{ll}a_{21} & a_{22} \\ a_{31} & a_{32}\end{array}\right|=a_{21} a_{32}-a_{31} a_{22}$.

Determinants Formula

The determinant formula for $2 \times 2$ matrices

$
\mathrm{A}=\left[\begin{array}{ll}
a_1 & a_2 \\
b_1 & b_2
\end{array}\right]
$

then $\operatorname{det} \mathrm{A}$ is :

$
|\mathrm{A}|=\left|\begin{array}{ll}
a_1 & a_2 \\
b_1 & b_2
\end{array}\right|=\mathrm{a}_1 \mathrm{b}_2-\mathrm{a}_2 \mathrm{b}_1
$

For a $3 \times 3$ matrix determinant can be calculated in the following way :

$
\text { let } \mathrm{A}=\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]
$

then we find $\operatorname{det} \mathrm{A}$ in following way

$
|A|=a_1\left(b_2 c_3-b_3 c_2\right)-a_2\left(b_1 c_3-c_1 b_3\right)+a_3\left(b_1 c_2-b_2 c_1\right)
$

This same process we follow to evaluate the determinant of the matrix of any order. Notice that we start the first term with the +ve sign then 2nd with the -ve sign and 3rd again +ve sign, this sign sequence is followed for any order of matrix.

How to Calculate Determinant of Matrices?

To calculate the determinant of a matrix, first ensure it is square (same number of rows and columns). For a 2x2 matrix, multiply the diagonal elements and subtract the product of the off-diagonal elements. Let us understand in detail:

Calculating the Determinant of $2 \times 2$ Matrix

Let $A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]$ be a matrix of order 2. Then the determinant of $A$ is defined as

$
|A|=\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right|=a_{11} \quad a_{22}-a_{21} a_{12}
$
Example: Let $|A|=$ $\left|\begin{array}{cc}2 & 4 \\ -1 & 2\end{array}\right|=(2 \times 2)-(-1 \times 4)=4+4=8$.

Calculating the Determinant of $3 \times 3$ Matrix

The determinant of matrix $
\mathrm{A}=\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]
$ is given by

$\qquad|\mathrm{A}|=\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=a_1\left|\begin{array}{ll}b_2 & c_2 \\ b_3 & c_3\end{array}\right|-b_1\left|\begin{array}{ll}a_2 & c_2 \\ a_3 & c_3\end{array}\right|+c_1\left|\begin{array}{ll}a_2 & b_2 \\ a_3 & b_3\end{array}\right|$

Example: $|A|= \left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$

$\begin{aligned} & =0\left|\begin{array}{cc}0 & \sin \beta \\ -\sin \beta & 0\end{array}\right|-\sin \alpha\left|\begin{array}{cc}-\sin \alpha & \sin \beta \\ \cos \alpha & 0\end{array}\right|-\cos \alpha\left|\begin{array}{cc}-\sin \alpha & 0 \\ \cos \alpha & -\sin \beta\end{array}\right| \\ & =0-\sin \alpha(0-\sin \beta \cos \alpha)-\cos \alpha(\sin \alpha \sin \beta-0) \\ & =\sin \alpha \sin \beta \cos \alpha-\cos \alpha \sin \alpha \sin \beta=0\end{aligned}$

Evaluation of determinants of matrices of order $3$ by the Sarrus rule,

Let $A=\left[a_{i j}\right]_{3 \times 3}=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$
Write the entries of Matrix $A$ as follows :

Then $|A|$ is computed as follows :

$
|A|=\left[a_{11} a_{22} a_{33}+a_{12} a_{23} a_{31}+a_{13} a_{21} a_{32}\right]-\left[a_{33} a_{21} a_{12}+a_{32} a_{23} a_{11}+a_{31} a_{22} a_{13}\right]
$

Notice that we start the first term with the +ve sign then 2nd with the -ve sign and 3rd again +ve sign, this sign sequence is followed for any order of matrix.

Example: $A=\left[\begin{array}{ccc}3 & 4 & 1 \\ 0 & -1 & 2 \\ 5 & -2 & 6\end{array}\right]$

$\begin{aligned}|A| & =[3(-1)(6)+4(2)(5)+1(0)(-2)]-[5(-1)(1)+(-2)(2) 3+6(0)(4)] \\ & =[-18+40+0]-[-5-12+0]=22+17=39\end{aligned}$

Multiplication of Determinants

There are two types of multiplication of determinants:

1) Multiplication of determinant by scalar quantity

2) Multiplication of determinant by another determinant

Multiplication of determinant by scalar quantity

If $A$ is a square matrix and $k$ is a scalar quantity then, $|\mathrm{kA}|=\mathrm{k}^{\mathrm{n}}|\mathrm{A}|$, where $n$ is the order of $A$

Multiplication of a determinant by another determinant

Determinant multiplication is a binary operation that produces a determinant from two determinants. For determinant multiplication, the order of both the determinants should be the same.

Multiplication of two determinants can be done by 4 methods, namely,

• row-by-row multiplication
• row-by-column multiplication
• column-by-row multiplication
• column-by-column multiplication

Multiplication of $2 \times 2$ Determinants

Let two determinants of second-order be

$\Delta_1=\left|\begin{array}{ll}a_1 & b_1 \\ a_2 & b_2\end{array}\right| \quad$ and $\quad \Delta_2=\left|\begin{array}{ll}\alpha_1 & \beta_1 \\ \alpha_2 & \beta_2\end{array}\right|$

We can multiply these by row-by-row or column-by-column or row-by-column or column-by-row

Row-by-row multiplication of these two determinants is given by

$\Delta_1 \times \Delta_2=\left|\begin{array}{ll}\left(a_1 \alpha_1+b_1 \alpha_2\right) & \left(a_1 \beta_1+b_1 \beta_2\right) \\ \left(a_2 \alpha_1+b_2 \alpha_2\right) & \left(a_2 \beta_1+b_2 \beta_2\right)\end{array}\right|$

Row-by-column multiplication of these two determinants is given by

$\Delta_1 \times \Delta_2=\left|\begin{array}{ll}\left(a_1 \alpha_1+b_1 \beta_1\right) & \left(a_1 \alpha_2+b_1 \beta_2\right) \\ \left(a_2 \alpha_1+b_2 \beta_1\right) & \left(a_2 \alpha_2+b_2 \beta_2\right)\end{array}\right|$

Column-by-row multiplication of these two determinants is given by

$\Delta_1 \times \Delta_2=\left|\begin{array}{ll}\left(a_1 \alpha_1+a_2 \alpha_2\right) & \left(b_1 \beta_1+b_2 \beta_2\right) \\ \left(a_1 \alpha_1+a_2 \alpha_2\right) & \left(b_1 \beta_1+b_2 \beta_2\right)\end{array}\right|$

Column-by-column multiplication of these two determinants is given by

$\Delta_1 \times \Delta_2=\left|\begin{array}{ll}\left(a_1 \alpha_1+a_2 \alpha_2\right) & \left(b_1 \beta_1+b_2 \beta_2\right) \\ \left(a_1 \alpha_1+a_2 \alpha_2\right) & \left(b_1 \beta_1+b_2 \beta_2\right)\end{array}\right|$

Multiplication of $3 \times 3$ Determinants

Let two determinants of third-order be

$\begin{equation}
\begin{aligned}
&\Delta_1=\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right| \text { and } \Delta_2=\left|\begin{array}{ccc}
\alpha_1 & \beta_1 & \gamma_1 \\
\alpha_2 & \beta_2 & \gamma_2 \\
\alpha_3 & \beta_3 & \gamma_3
\end{array}\right|
\end{aligned}
\end{equation}$

We can multiply these row-by-row or column-by-column or row-by-column or column-by-row

Row-by-row multiplication of these two determinants is given by

$\begin{equation}
\begin{aligned}
&\Delta_1 \times \Delta_2=\left|\begin{array}{lll}
a_1 \alpha_1+b_1 \beta_1+c_1 \gamma_1 & a_1 \alpha_2+b_1 \beta_2+c_1 \gamma_2 & a_1 \alpha_3+b_1 \beta_3+c_1 \gamma_3 \\
a_2 \alpha_1+b_2 \beta_1+c_2 \gamma_1 & a_2 \alpha_2+b_2 \beta_2+c_2 \gamma_2 & a_2 \alpha_3+b_2 \beta_3+c_2 \gamma_3 \\
a_3 \alpha_1+b_3 \beta_1+c_3 \gamma_1 & a_3 \alpha_2+b_3 \beta_2+c_3 \gamma_2 & a_3 \alpha_3+b_3 \beta_3+c_3 \gamma_3
\end{array}\right|
\end{aligned}
\end{equation}$

Row-by-column multiplication of these two determinants is given by

$\Delta_1 \times \Delta_2=\left|\begin{array}{lll}a_1 \alpha_1+b_1 \alpha_2+c_1 \alpha_3 & a_1 \beta_1+b_1 \beta_2+c_1 \beta_3 & a_1 \gamma_1+b_1 \gamma_2+c_1 \gamma_3 \\ a_2 \alpha_1+b_2 \alpha_2+c_2 \alpha_3 & a_2 \beta_1+b_2 \beta_2+c_2 \beta_3 & a_2 \gamma_1+b_2 \gamma_2+c_2 \gamma_3 \\ a_3 \alpha_1+b_3 \alpha_2+c_3 \alpha_3 & a_3 \beta_1+b_3 \beta_2+c_3 \beta_3 & a_3 \gamma_1+b_3 \gamma_2+c_3 \gamma_3\end{array}\right|$

Column-by-row multiplication of these two determinants is given by

$\Delta_1 \times \Delta_2=\left|\begin{array}{lll}a_1 \cdot \alpha_1+a_2 \cdot \beta_1+a_3 \cdot \gamma_1 & b_1 \cdot \alpha_1+b_2 \cdot \beta_1+b_3 \cdot \gamma_1 & c_1 \cdot \alpha_1+c_2 \cdot \beta_1+c_3 \cdot \gamma_1 \\ a_1 \cdot \alpha_2+a_2 \cdot \beta_2+a_3 \cdot \gamma_2 & b_1 \cdot \alpha_2+b_2 \cdot \beta_2+b_3 \cdot \gamma_2 & c_1 \cdot \alpha_2+c_2 \cdot \beta_2+c_3 \cdot \gamma_2 \\ a_1 \cdot \alpha_3+a_2 \cdot \beta_3+a_3 \cdot \gamma_3 & b_1 \cdot \alpha_3+b_2 \cdot \beta_3+b_3 \cdot \gamma_3 & c_1 \cdot \alpha_3+c_2 \cdot \beta_3+c_3 \cdot \gamma_3\end{array}\right|$

Column-by-column multiplication of these two determinants is given by

$\Delta_1 \times \Delta_2=\left|\begin{array}{lll}a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3 & b_1 \beta_1+b_2 \beta_2+b_3 \beta_3 & c_1 \gamma_1+c_2 \gamma_2+c_3 \gamma_3 \\ a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3 & b_1 \beta_1+b_2 \beta_2+b_3 \beta_3 & c_1 \gamma_1+c_2 \gamma_2+c_3 \gamma_3 \\ a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3 & b_1 \beta_1+b_2 \beta_2+b_3 \beta_3 & c_1 \gamma_1+c_2 \gamma_2+c_3 \gamma_3\end{array}\right|$

Properties of Determinants

The properties of determinants are given below:

If $A$ and $B$ are square matrices of same order:

• The determinant of the transpose of matrix $A$ is equal to the determinant of matrix $A$. $\operatorname{det}\left(A^{\prime}\right)=\operatorname{det} A$
• The product of the determinant of matrices $A B$ is equal to the product of the determinant of individual matrices. $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$
• The determinant of the skew-symmetric matrix of odd order is zero. if $A$ is the skew-symmetric matrix of odd order then $|A|=0$.
• The determinant of the skew-symmetric matrix of even order is a perfect square. if $A$ is a skew-symmetric matrix of even order then $|A|$ is a perfect square.
• $|k A|=k^n|A|$, where $n$ is the order of matrix $A$ and $k$ is a constant.
• $\left|A^n\right|=|A|^n$, where $n$ belongs to $N$.
• $|A B|=|B A|$

Important Formulae related to Determinants

List of topics related to determinants according to NCERT/JEE MAIN

This section covers all important topics on determinants as per NCERT syllabus and JEE MAIN.

Important Books and Resources for Determinants

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NCERT Resources

Access NCERT notes, solutions, and exemplar problems for determinants that simplify the study and help students prepare effectively.

NCERT Maths Notes for Class 12th Chapter 4 - Determinants
NCERT Maths Solutions for Class 12th Chapter 4 - Determinants
NCERT Maths Exemplar Solutions for Class 12th Chapter 4 - Determinants

NCERT Subjectwise Resources

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Practice Questions based on Determinants

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