q
Consider
the equations a_{1}x+b_{1}y = 0 and a_{2}x+b_{2}y
= 0. These give .

&⇒

&⇒ a_{1}b_{2} – a_{2}b_{1}
= 0

We express this eliminant as= 0.

q

q
A
determinant of order three consisting of 3 rows and 3 columns is written as and is equal to

q
The
numbers a_{i}, b_{i}, c_{i} ( i =1,2,3 ) are called
the elements of the determinant.

q The determinant
obtained by deleting the ith row and jth column is called the minor of element
at the ith row and the jth column. The cofactor of this element is (-1)^{i+j}
(minor). .

Note that : D = =
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.

q We can expand the determinant through any row or column. It means that we can write.

These results are true for determinants of any order.

The determinant remains unaltered if its rows are changed into columns and the columns into rows.

- If
all the elements of a row (or column) are zero, then the value of determinant

is zero. - If the elements of a row (column) are proportional (or identical) to the elements of any other row (column), then the determinant is zero.
- The interchange of any two rows (columns) of the determinant changes its sign.
- On
rolling over n rows the determinant value D
reduces to (–1)
^{n}D. . - If all the elements of a row (column) of a determinant are multiplied by a non-zero constant then the determinant gets multiplied by the same constant. .
- A
determinant remains unaltered under a column ( C
_{i}) operation of the form

C_{i}+ a C_{j}+ bC_{k}( j,k>i) or a row (R_{i}) operation of the form R_{i}+ a R_{j}+ bR_{k}( j,k>i). - If each element in any row (column) is the sum of r terms, then the determinant can be expressed as the sum of r determinants.
- If
determinant D = f(x) and f(a) =
0, then (x –a) is a factor of the determinant. In other word, if two rows (or
two columns) becomes proportional (identical) for x = a then (x - a) is a factor of determinant. In
general, if r rows become identical for x = a then (x - a)
^{r}^{-}^{1}is factor of the determinant. . - If
in a determinant (of order three or more) the elements in all the rows
(columns) are in A.P. with same or different common difference, the value of
the determinant

is zero. - The determinant value of an odd order skew symmetric determinant is always zero.

- It is important to know that all the properties applicable to rows are also equally applicable to columns but independently.
- Whenever rows are disturbed by applications of properties of determinants, at least one of the row shall remain in original shape. In other words all the rows shall not be disturbed at a time.
- It is always desirable to try to bring in as many zeros as possible in any row (or column) and then expand the determinant with respect to that row (column). Mere expansion from the outset should be avoided as far as possible.
- We can express a determinant as
- Where C
_{i}( i = 1,2, 3 ) are the columns and R_{j}( j=1,2,3) are the rows of the determinant.

Evaluate D only by using the
properties of determinant where

D =
.

Operating C_{1} → C_{1}
– C_{2} and C_{2} → C_{2} – C_{3}
, we get

D =

Operating C_{1} → C_{1}
– C_{2} and C_{3} → C_{3} – 10C_{2}
, we get

D =

Operating R_{1} → R_{1}
– R_{3} and R_{2} → R_{2} + 3R_{3}
, we get

D = = 0 (as first two rows are proportional).

**Without expanding to
any stage, prove that **

**D**** = **

D =

= D_{1} -D_{2}
(say)

D_{2} =

= = D_{1}

Hence D = D_{1} -D_{1} =
0

**Show that ****D**** = 0 if ****D**** = **

Operating C_{2}
→ C_{2} – (C_{1} + C_{3}),
we get

D = = 0

**Note: **Using the A.P. property one can
immediately write D = 0 directly .

**Using the factor
property of determinants show that**

**D**** = = k (a + b) (b + c) (c + a)****.
Evaluate k.**.

On checking, (with b = -a), we find that

On operating C_{1} → C_{1} + C_{2} &
C_{3} → C_{3} + C_{2}
we get

Taking 2 common from C_{1} and (a+c) from C_{3}

D = 2(a + c)

D = 2(a + c)

The operation R_{3}
→ R_{3} + R_{1}
and R_{2} → R_{2} + R_{3}
yields i.e. D= 0 .

Similarly it can be proved that (b + c) and (c + a) are factor of the determinant. On putting a = 1 , b = 1, c =1. .

R.H.S. = 8 k and L.H.S. = 32. Hence k = 4.