Determinants are one of the important concepts linked to matrices. Without understanding Determinants, learning about matrices is incomplete. The determinant of a matrix is determined by all of its components. The presence of a matrix's inverse is exactly proportional to the determinant's value. In Algebra, it is a highly helpful notion.
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Determinants are mathematical objects that may be used to solve and analyse systems of linear equations. Determinants are used in a variety of fields including engineering, science, economics, and social science. They may be used to solve linear equations, see how a linear transformation changes volume or area, and modify variables in integrals. Furthermore, we may think of determinants as a function with a square matrix as its input and an integer as its output. Let's have a look at the Determinants and its properties. This article is about the concept of determinants class 12.
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. A matrix is only a representation while the 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 $A_{13}=(-1)^{1+3} M_{13}=\left|\begin{array}{ll}a_{21} & a_{22} \\ a_{31} & a_{32}\end{array}\right|=a_{21} a_{32}-a_{31} a_{22}$.
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.
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 $|A|=$ $\left|\begin{array}{cc}2 & 4 \\ -1 & 2\end{array}\right|=(2 \times 2)-(-1 \times 4)=4+4=8$.
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.
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 determinants can be done by 4 methods, namely,
row-by-row multiplication
row-by-column multiplication
column-by-row multiplication
column-by-column multiplication
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
$\Delta_1 \times \Delta_2=\left|\begin{array}{ll}\left(a_1 \alpha_1+a_2 \alpha_2\right) & \left(b_1 \beta_1+b_2 \beta_2\right) \\ \left(a_1 \alpha_1+a_2 \alpha_2\right) & \left(b_1 \beta_1+b_2 \beta_2\right)\end{array}\right|$
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
$\Delta_1 \times \Delta_2=\left|\begin{array}{lll}a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3 & b_1 \beta_1+b_2 \beta_2+b_3 \beta_3 & c_1 \gamma_1+c_2 \gamma_2+c_3 \gamma_3 \\ a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3 & b_1 \beta_1+b_2 \beta_2+b_3 \beta_3 & c_1 \gamma_1+c_2 \gamma_2+c_3 \gamma_3 \\ a_1 \alpha_1+a_2 \alpha_2+a_3 \alpha_3 & b_1 \beta_1+b_2 \beta_2+b_3 \beta_3 & c_1 \gamma_1+c_2 \gamma_2+c_3 \gamma_3\end{array}\right|$
The properties of determinants are
If $A$ and $B$ are square matrices of same order:
Adjoint And Inverse Of A Matrix | Cramer's Rule |
Inverse Matrix | Homogeneous System Of Linear Equations |
System Of Linear Equations | Solving Linear Equations Using Matrix |
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.
Start preparing by understanding and practising 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.
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.
A determinant is a real integer that is not a matrix in most cases. A determinant, on the other hand, can be a negative number. Most notably, it has no resemblance to absolute value other than the usage of vertical lines.
If the determinant of a square matrix n n A is 0, then A is not invertible in general. Furthermore, if a matrix's determinant is not zero, the linear system it represents is linearly independent. When a matrix's determinant is 0, the rows become linearly dependent vectors, and the columns become linearly dependent vectors.
It's a method for solving systems of equations with the same number of equations as variables that use determinants. It also takes into account a two-variable system of two linear equations.
A positive definite matrix's determinant must always be positive. As a result, a non-singular matrix is always a positive definite matrix. The inverse of a positive definite matrix is also a positive definite matrix.
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