DETERMINANT
1. 3.
DETERMINANT OF A SQUARE MATRIX OF ORDER TWO AND THREE
PROPERTIES OF DETERMINANTS
a1 b1
Expansion of two order: = a1b2 − b1a2
a2 b2 (i) The value of a determinant remain
a1 b1 c1 interchanged.
b2 c2 a c2 a b2 (ii) If any two rows (or columns) of a
Expansion of three order: a2 b2 c2 = a1 − b1 2 + c1 2
b3 c3 a3 c3 a3 b3 of determinant changes.
a3 b3 c3
(iii) If any two rows (or columns) of a
of determinant is zero.
2. (iv) If each element of a row (or a col
constant k, then its value gets mu
SARRUS RULE a1 b1 c1 (v) If some or all the elements of a ro
∆ = a 2 b2 c2 expressed as a sum of two (or m
a 3 b 3 c3 expressed as a sum of two (or mo
a1 b1 c1 (vi) If the equimultiples of correspond
added to each element of any row
a2 b2 c2 value of the determinant remains
a3 b3 c3 (vii) |AT |=|A|, where AT= transpose of A
a3b 2c1 a1b 2c3
+ a1 b1 c1 + (viii) If A = [aij]3×3, then |kA| = k3 |A|.
a1b 3c2 a2b 3c1
+ a2 b2 c2 + (ix) The determinant of the product o
a2b 1c3 a3b 1c2 respective determinants, i.e., |AB|=
N P of same order
(x) a1 b1 c1 a1 0 0 a1 0
⇒∆=P–N
0 b2 c2 = a 2 b2 0 = 0 b2
0 0 c3 a 3 b 3 c3 0 0
4.
USE OF DETERMINANTS IN CO-ORDINATE GEOMETRY
x y 1 (iv) If three lines arx + bry + cr = 0 are
1 1 1
(i) Area of triangle, whose vertices are (xr, yr)∆ = x 2 y2 1
2x y 1
3 3
(v) If ax2 + 2hxy + by2 + 2gx + 2fy + c
(ii) If arx + bry + cr = 0 are the sides of a triangle, then the area =
a h g
a1 b1 c1
2 line then h b f = 0
1 g f c
a 2 b2 c2 c1, c2, c3 are cofactors of c1, c2, c3.
2c1c 2 c3 a b c
3 3 3
(vi) The equation of circle through t
(iii) Equation of a straight line passing through two points
x 2 + y2 x y 1
x y 1
x12 + y12 x1 y1 1
(x1, y1) & (x2, y2) is x1 y1 1 = 0 = 0.
x 2 y2 1 x 22 + y 22 x 2 y 2 1
x 32 + y32 x 3 y3 1
5. 6.
MINOR AND COFACTOR OF AN ELEMENT OF A DETERMINANT ADJOINT OF A MATRIX
Minor: The determinant that is left by cancelling the row and a11 a12 a13
column intersecting at a particular element of a determinant is
1. 3.
DETERMINANT OF A SQUARE MATRIX OF ORDER TWO AND THREE
PROPERTIES OF DETERMINANTS
a1 b1
Expansion of two order: = a1b2 − b1a2
a2 b2 (i) The value of a determinant remain
a1 b1 c1 interchanged.
b2 c2 a c2 a b2 (ii) If any two rows (or columns) of a
Expansion of three order: a2 b2 c2 = a1 − b1 2 + c1 2
b3 c3 a3 c3 a3 b3 of determinant changes.
a3 b3 c3
(iii) If any two rows (or columns) of a
of determinant is zero.
2. (iv) If each element of a row (or a col
constant k, then its value gets mu
SARRUS RULE a1 b1 c1 (v) If some or all the elements of a ro
∆ = a 2 b2 c2 expressed as a sum of two (or m
a 3 b 3 c3 expressed as a sum of two (or mo
a1 b1 c1 (vi) If the equimultiples of correspond
added to each element of any row
a2 b2 c2 value of the determinant remains
a3 b3 c3 (vii) |AT |=|A|, where AT= transpose of A
a3b 2c1 a1b 2c3
+ a1 b1 c1 + (viii) If A = [aij]3×3, then |kA| = k3 |A|.
a1b 3c2 a2b 3c1
+ a2 b2 c2 + (ix) The determinant of the product o
a2b 1c3 a3b 1c2 respective determinants, i.e., |AB|=
N P of same order
(x) a1 b1 c1 a1 0 0 a1 0
⇒∆=P–N
0 b2 c2 = a 2 b2 0 = 0 b2
0 0 c3 a 3 b 3 c3 0 0
4.
USE OF DETERMINANTS IN CO-ORDINATE GEOMETRY
x y 1 (iv) If three lines arx + bry + cr = 0 are
1 1 1
(i) Area of triangle, whose vertices are (xr, yr)∆ = x 2 y2 1
2x y 1
3 3
(v) If ax2 + 2hxy + by2 + 2gx + 2fy + c
(ii) If arx + bry + cr = 0 are the sides of a triangle, then the area =
a h g
a1 b1 c1
2 line then h b f = 0
1 g f c
a 2 b2 c2 c1, c2, c3 are cofactors of c1, c2, c3.
2c1c 2 c3 a b c
3 3 3
(vi) The equation of circle through t
(iii) Equation of a straight line passing through two points
x 2 + y2 x y 1
x y 1
x12 + y12 x1 y1 1
(x1, y1) & (x2, y2) is x1 y1 1 = 0 = 0.
x 2 y2 1 x 22 + y 22 x 2 y 2 1
x 32 + y32 x 3 y3 1
5. 6.
MINOR AND COFACTOR OF AN ELEMENT OF A DETERMINANT ADJOINT OF A MATRIX
Minor: The determinant that is left by cancelling the row and a11 a12 a13
column intersecting at a particular element of a determinant is
