Determinants

Minors & co-factors
Cofactor = (-1)i+j Minor

Expansion
Only square determinants can be solved.

For determinants having more than three rows and columns, Det = aA - bB + cC - dD ........

where a,b,c,d are the first row elements and A,B,C,D are their corresponding minors

A determinant having all elements of a row or column zero, has value zero.

Reduction
A determinant evaluation can be made easy by reducing the determinant into a unit matrix.

Properties
1) Det = aA - bB + cC - dD ........

where a,b,c,d are the first row elements or column elements and A,B,C,D are their corresponding minors

2) One row or column can be subtracted or added by another row or column, respectively ; the value remains same.

3) An entire row or column can be multiplied by a scalar ; the value remains same.

Sum of Determinants
Corresponding elements of determinants can be added directly. Thus, both the determinants should be of the same order.

Product of Determinants
Multiplication of Determinants is not commutative.

Derivative of Determinant
Der = Sum of Determinants with first row derivative then second row derivative .... n row derivative

Cramer's Rule
Determinants can be used to find the solution of linear equations

Number of Solutions of a Linear Equation
Δ =/ 0 ; then one unique solution

Δ = 0 & Δx, Δy , Δzis equal to zero ; then infinite solutions

Δ=0, but at least one of Δx , Δy , Δzis not equal to zero ; then equation has no solutions.

Consistency of Equation
Consider a determinant of co-efficients and constants of three linear equations.

if Det = 0, then the equations are consistent ; i.. at least one set of solution exists.

Area
Area of Triangle = 1/2 | |

Area of Quadrilateral : Divide quadrilateral into two triangles and use the above formula.