__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|**

**Determinants is a number associated with the square matrix.**

**For Example: ****Let a matrix A of order 2 x 2 is given by **

then **Let a matrix of order 3 x 3 is given by : **

**Determinant of matrix A is denoted by |A| and is given by **

**Expand it along the first row **

**|A| = 3(25 - 4) - 4(- 10 + 12) + 5(2 - 15)**

**|A| = 3 × 21 - 4 × 2 + 5 × - 13**

**|A| = 63 - 8 - 65 = -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}

**
**

**Shortcut Method of finding the Adjoint**

Let us suppose a matrix A given by

Write first two columns to the extreem right of this matrix without brackets we get

Wright first two rows again below the third row we get

Now leave first row and first column we get

**Now cross multiply the numbers as shown in the figure**

**Adjoint of A is given by **

**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: ****\[ A^{-1}=\frac{1}{\left | A \right |}\times AdjointA\]**** **** **

**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

__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 (no solution) and the system of equations is called inconsistent.**

**2. If |A| = 0 and (adj. A)B = O, then the system of equations have infinitly many solutions and the system is called consistent.**

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

**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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