## Determinants Definitions & Properties

DefinitionsConsider the equations a

_{1}x + b_{1}y = 0, a_{2}x + b_{2}y = 0. These give -a_{1}/b_{1 = }y/x_{ = }-_{ }a_{2}/b_{2}=> a_{1}/b_{1}= a_{2}/b_{2}=> a

_{1}b_{2}- a_{2}b_{1}= 0.We express this eliminant as = 0.

The expression is called a determinant of order two, and equals a

_{1}b_{2}-a_{2}b_{1}.A determinant of order three consisting of 3 rows and 3 columns is written as and is equal to a

_{1}= a_{1}(b_{2}c_{3}-c_{2}b_{3})-b_{1}(a_{2}c_{3}-c

_{2}a_{3})+c_{1}(a_{2}b_{3}-b_{2}a_{3}).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 = =a_{1}A_{1 }+ b_{1}B_{1}+ c_{1}C_{1}where A

_{1}, B_{1}and C_{1}are the cofactors of a_{1}, b_{1}and c_{1}respectively.We can expand the determinant through any row or column. It means we can write Δ = a

_{2}A_{2}+ b_{2}B_{2}+ c_{2}C_{2}or Δ = a_{1}A_{1}+ a_{2}A_{2}+ a_{3}A_{3}etc.Also a

_{1}A_{2}+ b_{1}B_{2}+ c_{1}C_{2}= 0=> a

_{i}A_{j}+ b_{i}B_{j}+ c_{i}C_{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 aR

_{1}, bR_{2}, cR_{3}and then take abc common out of C_{3}=> L.H.S. = .

Interchange C

_{1}and C_{3}and then C_{2}and C_{3}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 C

_{1}-> C_{1}+ C_{2}+ C_{3}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.

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