Definitions

Consider the equations a₁x + b₁y = 0, a₂x + b₂y = 0. These give -a₁/b_{1 =} y/x ₌ - a₂/b₂ => a₁/b₁ = a₂/b₂

=> a₁b₂ - a₂b₁ = 0.

We express this eliminant as = 0.

The expression is called a determinant of order two, and equals a₁b₂-a₂b₁.

A determinant of order three consisting of 3 rows and 3 columns is written as and is equal to a₁ = a₁(b₂c₃-c₂b₃)-b₁(a₂c₃-c

₂a₃)+c₁(a₂b₃-b₂a₃).

The numbers a_i, b_i, c_i (i = 1 2, 3,) are called the elements of the determinant.

The determinant, obtained by deleting the ith row and the jth column is called the minor of the element at ith row and jth column. The cofactor of this element is (-1)^i+j (minor). Note that: Δ = =a₁A₁ + b₁B₁ + c₁C₁

where A₁, B₁ and C₁ are the cofactors of a₁, b₁ and c₁ respectively.

We can expand the determinant through any row or column. It means we can write Δ = a₂A₂ + b₂B₂ + c₂C₂ or Δ = a₁A₁ + a₂A₂ + a₃A₃ etc.

Also a₁A₂ + b₁B₂ + c₁C₂ = 0

=> a_iA_j + b_iB_j + c_iC_j       = Δ   if      i = j,

                                   = 0   if      i ≠ j.

These results are true for determinants of any order.

Properties of Determinants

(i)     If rows be changed into columns and columns into the rows, then the values of the determinant remains unaltered.

(ii)    If any two row (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.

(iii)    If two rows (or two columns) in a determinant have corresponding elements that are equal, the value of determinant is equal to zero.

(iv)   If each element in a row (or column) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants.

Illustration:

        Without expanding, show that

Solution:

        L.H.S. =

        On the second determinant operate aR₁, bR₂, cR₃ and then take abc common out of C₃

        => L.H.S. = .

        Interchange C₁ and C₃ and then C₂ and C₃

        So that L.H.S. = .

(vi)   If to each element of a line (row or column) of a determinant be added the equimultiples of the corresponding elements of one or more parallel lines, the determinant remains unaltered

i.e. .

Illustration:

        Evaluate where ω is cube root of unit.

Solution:

        Applying C₁ -> C₁ + C₂ + C₃ we get

        = 0.

(vii)   If each element in any row (or any column) of determinant is zero, then the value of determinant is equal to zero.

(viii)  If a determinant D vanishes for x = a, then (x - a) is a factor of D, In other words, if two rows (or two columns) become identical for x = a, then (x-a) is a factor of D.

        In general, if r rows (or r columns) become identical when a is substituted for x, then (x-)^r-1 is a factor of D.