Consider the system of two homogeneous linear equations
`a_1x + b_1y =0..................(i)`
`a_2x + b_2y=0.................(ii)`
in the two variables x and y. From these equations, we obtain
`-a_1/b_1 = y/x = -a_2/b_2 => a_1/b_1 = a_2/b_2`
`=> a_1b_2 - a_2b_1 =0`
The result `a_1 b_2 - a_2b_2` is represented by
`|(a_1,b_1),(a_2,b_2)|`
which is known as determinant of order two. The quantities `a_1, b_1, a_2` and `b_2` are
called constituents or elements of the determinant and `a_1 b_2 - a_2b_1` is called its value.
The horizontal lines are called rows and vertical lines are called columns.
Here, this determinant consists two rows and two columns.
For example, The value of the determinant
`|(2,3),(4,-5)| = 2 xx (-5) -3 xx 4 = -10 -12 = -22`
Now, let us consider the system of three homogeneous linear equations
`a_1x + b_1y + c_1z = 0....................(i)`
`a_2 x + b_2y + c_2 z = 0.................(ii)`
`a_:3x + b_3y + c_3z = 0...............(iii)`
On solving Eqs. (ii) and (iii) for x, y and z by cross-multiplication, we get
`x/(b_2c_3- b_3c_2) = y/(c_2a_3 -c_3a_2)= z/(a_2b_3-a_3b_2) = k`
`=> x = k(b_2c_3-b_3c_2), y= k(c_2a_3-c_3a_2)`
and `z = k(a_2b_3 - a_3b_2)`
On putting thEJse values of x, y and z in Eq. (i), we get
`a_1(b_2c_3 - b_3c_2) + b_2(c_1a_3 - c_3a_2) + c_1(a_2b_3- a_3b_2) = 0`
or `a_1(b_2c_3 - b_3c_2) + b_2(c_3a_2 - c_2a_3) + c_1(a_2b_3- a_3b_2) = 0....................(iv)`
or `a_1|(b_2,c_2),(b_3,c_3) - b_1 |(c_2,a_2),(c_3,a_3)| c_1 |(a_2,b_2),(a_3,b_3)|=0 ....................(v)`
Usually this is written as
`|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)| =0`
Here, the expression `|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)| ` consisting of three rows and three columns,
is called determinant of order three.
the quantities `a_1, b_1, c_1_, a_2 , b_2, c_2 ,a_3,b_3 ` and `c_3` are called constituents or
elements of the determinant.e.
`text(Note :)`
1. A determinant is generally denoted by D or `Delta` .
2. A determinant of the nth order consists of n rows and n columns and its
expansion contains n ! terms.
3. A determinant of the nth order consists of n rows and n columns.
:. Number of constituents in determinant `= n^2`
4. In a determinant the horizontal lines counting from top to bottom 1st, 2nd,
3rd, .. . respectively, known as rows and denoted by `R_1, R_2 , R_3, .. .` and
vertical lines from left to right 1st, 2nd, 3rd, ... respectively, known as
columns and denoted by `c_1 ,c_2, c_3 ....`
5. Shape of every determinant is square.
Consider the system of two homogeneous linear equations
`a_1x + b_1y =0..................(i)`
`a_2x + b_2y=0.................(ii)`
in the two variables x and y. From these equations, we obtain
`-a_1/b_1 = y/x = -a_2/b_2 => a_1/b_1 = a_2/b_2`
`=> a_1b_2 - a_2b_1 =0`
The result `a_1 b_2 - a_2b_2` is represented by
`|(a_1,b_1),(a_2,b_2)|`
which is known as determinant of order two. The quantities `a_1, b_1, a_2` and `b_2` are
called constituents or elements of the determinant and `a_1 b_2 - a_2b_1` is called its value.
The horizontal lines are called rows and vertical lines are called columns.
Here, this determinant consists two rows and two columns.
For example, The value of the determinant
`|(2,3),(4,-5)| = 2 xx (-5) -3 xx 4 = -10 -12 = -22`
Now, let us consider the system of three homogeneous linear equations
`a_1x + b_1y + c_1z = 0....................(i)`
`a_2 x + b_2y + c_2 z = 0.................(ii)`
`a_:3x + b_3y + c_3z = 0...............(iii)`
On solving Eqs. (ii) and (iii) for x, y and z by cross-multiplication, we get
`x/(b_2c_3- b_3c_2) = y/(c_2a_3 -c_3a_2)= z/(a_2b_3-a_3b_2) = k`
`=> x = k(b_2c_3-b_3c_2), y= k(c_2a_3-c_3a_2)`
and `z = k(a_2b_3 - a_3b_2)`
On putting thEJse values of x, y and z in Eq. (i), we get
`a_1(b_2c_3 - b_3c_2) + b_2(c_1a_3 - c_3a_2) + c_1(a_2b_3- a_3b_2) = 0`
or `a_1(b_2c_3 - b_3c_2) + b_2(c_3a_2 - c_2a_3) + c_1(a_2b_3- a_3b_2) = 0....................(iv)`
or `a_1|(b_2,c_2),(b_3,c_3) - b_1 |(c_2,a_2),(c_3,a_3)| c_1 |(a_2,b_2),(a_3,b_3)|=0 ....................(v)`
Usually this is written as
`|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)| =0`
Here, the expression `|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)| ` consisting of three rows and three columns,
is called determinant of order three.
the quantities `a_1, b_1, c_1_, a_2 , b_2, c_2 ,a_3,b_3 ` and `c_3` are called constituents or
elements of the determinant.e.
`text(Note :)`
1. A determinant is generally denoted by D or `Delta` .
2. A determinant of the nth order consists of n rows and n columns and its
expansion contains n ! terms.
3. A determinant of the nth order consists of n rows and n columns.
:. Number of constituents in determinant `= n^2`
4. In a determinant the horizontal lines counting from top to bottom 1st, 2nd,
3rd, .. . respectively, known as rows and denoted by `R_1, R_2 , R_3, .. .` and
vertical lines from left to right 1st, 2nd, 3rd, ... respectively, known as
columns and denoted by `c_1 ,c_2, c_3 ....`
5. Shape of every determinant is square.