__Class XII : Chapter : 4 : Determinants__

*Formulas and basic concepts based on the determinants chapter 4 class XII, properties of determinants, method of finding the solutions of linear equations by matrix method.*

**
Determinant:**** **
**To every square matrix A = ****[a**_{ij}] **of
order n, we can associate a number (real or complex) called determinant of the
square matrix A. ****Determinant of matrix [A] is denoted by |A|**

**For Example: ****Let a matrix A of order 2 x 2 is given by \[ A=\; \left [ \begin{matrix} a &b \\ c & d
\end{matrix} \right ]\]\[Then \; \; \left |A \right | =\left |
\begin{matrix} a &b \\ c & d \end{matrix} \right |= ad-cb\]****Let a matrix of order 3 x 3 is given by : \[A=\begin{bmatrix} 3 &4 &5 \\ -2& 5 & -4\\ 3 & -1 & 5 \end{bmatrix}\]**

**Determinant of matrix A is denoted by |A| and is given by \[|A|=\begin{vmatrix} 3 &4 &5 \\ -2& 5 & -4\\ 3 & -1 & 5\end{vmatrix}\]****Expand it along the first row \[|A|= 3\begin{vmatrix} 5 &-4 \\ -1&5 \end{vmatrix}-4\begin{vmatrix} -2 &-4 \\ 3&5 \end{vmatrix}+5\begin{vmatrix} -2 &5 \\ 3 &-1 \end{vmatrix}\] \[|A|=3(25-4)-4(-10+12)+5(2-15)\] \[|A|=3\times 21-4\times 2+5\times -13\] \[|A|=63-8-65=-8-2=-10\]**

**Note:**

**1. For matrix A, |A| is read as
determinant of A and not modulus of A.**

**2. Only square matrix have
determinant.**

**3. For easier calculation to find
the determinants of matrix of order 2 x 2, we shall expand the determinant
along that row or column which contains maximum number of zeroes.**

**4. ****Rule to write the sign (+ or - ) of the position of the element while finding the determinant.**

**Position of a**_{11} is (-1)^{1+1} = (-1)^{2}
= +ve

**Position of a**_{12} is (-1)^{1+2} = (-1)^{3}
= - ve

**Position of a**_{13} is (-1)^{1+3} = (-1)^{4}
= +ve

**Position of a**_{21} is (-1)^{2+1} = (-1)^{3}
= - ve

**Position of a**_{22} is (-1)^{2+2} = (-1)^{4}
= +ve

**……………….. and so on**

**For 2 x 2 Matrix we can take sigh as follows \[\begin{bmatrix} + &- \\ -&+ \end{bmatrix}\]****For 3 x 3 Matrix we can take sign as follows \[\begin{bmatrix} + & - &+ \\ -& + & -\\ +& - &+ \end{bmatrix}\]**

__Important Properties of
determinants :__

**Property 1:**** The value of the determinant remains unchanged if its rows
and columns are interchanged. |A| = |A'|**

**Property 2:**** If any two rows (or columns) of a determinant are
interchanged then the sign of the determinant is changes. **

**Property 3 :**** If in a matrix any two rows (or columns) are identical then
the determinant of the matrix is zero.** \[|A|=\begin{vmatrix} a &b &c \\ a &b & c\\ 3& 5 & 7 \end{vmatrix}=0\]

**Property 4:**** If each element of a row (or column) of a determinant
is multiplied by a constant k, then the value of the determinant gets
multiplied by k. or multiplying a determinant by any constant k(say)
means, multiply each element of only one row (or column) by k.**

**Property 5:**** If some or all elements of a row or column of a determinant
are expressed as sum of two or more terms, then the determinant can be
expressed as the sum of two or more determinants.**

**\[If\; A=\left | \begin{matrix} a+b
&c+d &e+f \\ 1&2 &3 \\ 4&5 &6 \end{matrix} \right | \;
then\; A\; can\; be\; written\; as\]\[A=\left | \begin{matrix} a & c&e
\\ 1& 2 & 3\\ 4&5 & 6 \end{matrix} \right |+\left |
\begin{matrix} b &d &f \\ 1 &2 &3 \\ 4 &5 &6
\end{matrix} \right |\]Same property can be applied in columns.**

**Property 6:****
If each element of a row (or column) are multiplied by the same number and is
added to the corresponding elements of the other row (or column) , then
the value of the determinants remain unchanged. **

**This means that the value of
determinant remain unchanged if we apply the following operations**

**\[R_{i}\rightarrow R_{i}+kR_{j}\;
\;Or \; \; R_{i}\rightarrow R_{i}-kR_{j}\; \; Or\]\[C_{i}\rightarrow
C_{i}+kC_{j}\; \;Or \; \; C_{i}\rightarrow C_{i}-kC_{j}\; \; \]**

**Property 7:**** If A is a matrix of order n x n then |kA| = k**^{n}|A|

**Similarly If A is a
matrix of order 3 x 3 then |kA| = k**^{3}|A| or
**If A = kB, where A and B are two square matrices of order n, **

**Then |A| = k**^{n} |B|

**Property 8: If all elements of a row or a column of a matrix are zero then the value of that determinant is also become zero. **\[|A|=\begin{vmatrix} 0 &0 &0 \\ a &b & c\\ 3& 5 & 7 \end{vmatrix}=0\]

__Method of finding the area of triangle with the help of Determinants__

** If A is a matrix and
A' is the transpose of the matrix then |A'| = |A|**

Area of Triangle ABC with vertices
(x_{1}, y_{1}), (x_{2}, y_{2}) , (x_{3},
y_{3})
**\[=\frac{1}{2}\left | \begin{matrix}
x_{1} &y_{1} &1 \\ x_{2}& y_{2} &1 \\ x_{3}&y_{3} &1
\end{matrix} \right |\]**

**\[=\frac{1}{2}\left | x_{1}\left (
y_{2}-y_{3}\right )+x_{2}\left ( y_{3}-y_{1} \right )+x_{3}\left ( y_{1}-y_{2}
\right ) \right |\]**

**Note :**

**1. Area is a positive quantity, so
we always take the absolute value of the determinant for finding area.**

**2. When area is given, then use both
positive and negative values of the determinants for calculation.**

**3. The area of the triangle formed
by joining three collinear points is always zero.**

**Minor:**

** Minor of an element a**_{ij} of
a determinant is the determinant obtained by deleting its i^{th} row
and j^{th} column in which element a_{ij} lies. Minor
of element a_{ij} is denoted by M_{ij}.

**Co-factors :**

**Co-factors of an element a**_{ij},
is denoted by A_{ij} is defined by A_{ij} = (-1)^{i+j} M_{ij},
where M_{ij} is minor of a_{ij}.

**Note:**

**1. Sum of the product of the
elements of any row (or column) with their corresponding co-factors, gives
determinant.**

**|A|
= a**_{11}A_{11} + a_{12}A_{12 }+ a_{13}A_{13}

**2. If elements of any row (or
column) are multiplied with the co-factors of any other row (or column) then
their sum is zero.**

**
a**_{11}A_{12} + a_{12}A_{13 }+
a_{13}A_{11} = 0 ** **_{ }

_{}

__Adjoint and Inverse of Matrix__

**Adjoint of a matrix :**** **

**Adjoint of a matrix is defined as
the transpose of the co-factor matrix.**

**If A is any square matrix then
: A(adj A) = (adj A) A = |A|I**

**A square matrix A is said to be singular if |A| = 0**

**
**

**A square matrix A is said to be non- singular if |A| ≠ 0**

**
**

**If A is a square matrix of order n then |adj.A| = |A|**^{n-1}

**
**
**If AB = BA = I then A and B are said
to be the inverse of each other. or A**^{-1} = B or B^{-1} =
A and (A^{-1})^{-1} = A

**Inverse of the matrix:**

**Inverse of matrix A is calculated by using the formula :\[ A^{-1}=\frac{1}{\left | A \right |}\times AdjointA\]**

**Method of finding Inverse of a Matrix**

**Step 1 : Name the
given matrix A (say)**

**Step 2 : Find determinant of
matrix A or find |A|**

**Step 3 : If |A| = 0, then A
is a singular matrix and ****A**^{-1}** does not exist.** **If |A| ≠0, then A is a non
singular matrix and ****A**^{-1}** exists.**

**Step 4 : If ****A**^{-1}** exists then find cofactor matrix of
A.**

**Step 5 : From co-factor
matrix find adjoint A.**

**Step 6 : Find the inverse of
Matrix A by using the formula: ****\[ A^{-1}=\frac{1}{\left | A \right |}\times AdjointA\]**** **** **

__Applications of
Determinants and Matrices__

**If given system of linear equations
either have one or more solutions then the system is called consistent. If |A| ****≠ 0, then system of equations have unique solution, and the system is called consistent.**

**System of linear equations is said
to be inconsistent if its solution does not exist.**

**If A is a singular matrix, then |A| = 0. In this case , we calculate (adj. A)B and following cases arise.**

**1. If |A| = 0 and (adj. A)B ≠ O, then solution does not exist and the system of equations is called inconsistent.**

**2. If |A| = 0 and (adj. A)B = O, then the system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.**

**a) If system of equations have many solution then the system is consistent. **

**b) If system have no solution then it is inconsistent.**

**Flow Diagram to understand consistency or inconsistency**

__Solution of system of
linear equation using matrix method__

**Let the given system of linear
equations is**

**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}

**This system of equations can be
written in matrix form as **

**\[A=\left [\begin{matrix} a_{1} &b_{1} &c_{1} \\ a_{2}& b_{2} & c_{2}\\ a_{3}& b_{3} & c_{3} \end{matrix} \right ],\; \; X= \left [\begin{matrix} x\\y \\z \end{matrix} \right ],\; \; B= \left [\begin{matrix} d_{1}\\d_{2} \\d_{3} \end{matrix} \right ]\]**

**In Matrix form system of equations can be written as **

**AX = B ⇒ X = ****A**^{-1}** B **

**Steps to follow for solving system of linear equations**

**Step 1 : Write the given equations in the form AX = B, where A is the coefficient matrix, X is the variable matrix and B is the matrix of constant terms.**

**Step 2 : Find |A| , ****If system of equations is non-singular i.e. |A| ≠ 0, system of equations have unique solution.**

**Step 3 : If ****|A| ≠ 0, then find ****A**^{-1}** by using the steps discussed above.**

**Step 4 : Find the value of the variables x, y, z by using the formula : ****X = A**^{-1}
B

********************************************************************

**Cramer's Rule: (Only for Applied Maths Students)****Let system of linear equations given by**

**\[a_{1}x+b_{1}y=d_{1}\\a_{2}x+b_{2}y=d_{2}\]**

**Solution of these equations by using cramer's rule is given by**

**\[x=\frac{\Delta _{1}}{\Delta },\;\; y=\frac{\Delta _{2}}{\Delta }\]**

**\[Where\; \Delta =\begin{vmatrix} a_{1}& b_{1}\\ a_{2}& b_{2} \end{vmatrix},\; \; \Delta _{1}=\begin{vmatrix} d_{1} & b_{1}\\ d_{2} &b_{2} \end{vmatrix},\; \; \Delta _{2}=\begin{vmatrix} a_{1} &d_{1} \\ a_{2} & d_{2} \end{vmatrix}\]\[Provided:\; \; \Delta \neq 0\]**

*******************************************************************

**Let system of equations are given by **

**\[a_{1}x+b_{1}y+c_{1}z=d_{1}\]\[a_{2}x+b_{2}y+c_{2}z=d_{2}\]\[a_{3}x +b_{3}y +c_{3}z = d_{3}\]**

**Solution of system of equations is given by **

**\[x=\frac{\Delta _{1}}{\Delta },\;\; y=\frac{\Delta _{2}}{\Delta },\;\; z=\frac{\Delta _{3}}{\Delta }\]**

**\[Where\; \; \Delta =\begin{vmatrix} a_{1} &b_{1} & c_{1}\\ a_{2} & b_{2} &c_{2} \\ a_{3}& b_{3} & c_{3} \end{vmatrix},\; \; \Delta_{1} =\begin{vmatrix} d_{1} &b_{1} & c_{1}\\ d_{2} & b_{2} &c_{2} \\ d_{3}& b_{3} & c_{3} \end{vmatrix}\]**

**\[Where\; \; \Delta_{2} =\begin{vmatrix} a_{1} &d_{1} & c_{1}\\ a_{2} & d_{2} &c_{2} \\ a_{3}& d_{3} & c_{3} \end{vmatrix},\; \; \Delta_{3} =\begin{vmatrix} a_{1} &b_{1} & d_{1}\\ a_{2} & b_{2} &d_{2} \\ a_{3}& b_{3} & d_{3} \end{vmatrix}\]\[Provided:\; \; \Delta \neq 0\]**

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