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 determinants 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?

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 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 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 detrminants 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

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 $N$.
• $|A B|=|B A|$

List of Topics According to NCERT/JEE MAIN

Importance of Determinants Class 12

Determinants have a significant weighting in the IIT JEE test, which is a national level exam for 12th grade students that aids in admission to the country's top engineering universities. It is one of the most difficult exams in the country, and it has a significant impact on students' futures. Several students begin studying as early as Class 11 in order to pass this test. When it comes to math, the significance of these chapters cannot be overstated due to their great weightage. You may begin and continue your studies with the standard books and these revision notes, which will ensure that you do not miss any crucial ideas and can be used to revise before any test or actual examination.

How to Study Determinants?

Start preparing by understanding and practicing to find the value of the determinants. Try to be clear on concepts like minors and cofactors of determinants and their properties. Practice many problems from each topic for better understanding.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

NCERT Notes Subject wise link:

• NCERT Notes for class 12 Physics.
• NCERT Notes for class 12 Chemistry.
• NCERT Notes for class 12 Mathematics.
• NCERT Notes for class 12 Biology

Important Books for Determinants

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Arihant Algebra Textbook by SK Goyal or Cengage Algebra Textbook by G. Tewani but make sure you follow any one of these not all. Determinants are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.

NCERT Solutions Subject wise link:

• NCERT solutions for class 12 Physics.
• NCERT solutions for class 12 Chemistry.
• NCERT solutions for class 12 Mathematics.
• NCERT solutions for class 12 Biology.

NCERT Exemplar Solutions Subject wise link:

• NCERT exemplar solutions for class 12 Physics.
• NCERT exemplar solutions for class 12 Chemistry.
• NCERT exemplar solutions for class 12 Mathematics.
• NCERT exemplar solutions for class 12 Biology.