Question

Given the adj of a matrix find the determinant

Answer 1

we have two types of matrices.
1)square matrix : a b
c d
then its adj is written as -a c
b -d.
(here interchange b and c and keep negative signs to a and c)
eg: 1 2 then its adj become as -1 3
3 4 2 -4

it's det is found as below :

a b then its det is ab - cd
c d
(multiply opp terms and keep pop sign).

eg: 1 2. then it's det is 1(4) -2(3) =4-6=-2
3 4
1. 2 then its det is. 1(4) -(-6) =4 + 6= 10
-3 4

2) in 3 into 3 matrix. :

a b c.
d e f
g h i then its det is. :
a (e(i)-f(h)) - b (d(i)-f(g) ) + c (d(h) -g(e))
( here leave the row and column of number.)
eg: 1 2 3
4. 5 6
3. 2 1
then det is
1 (5(1)-2(6)) - 2(4(1)-6(3)) + 3(4 (2)-5(3))
=1(5-12)-2(4-18)+3 (8-15)
=1(-7)-2(-14)+3(-7) = -7+28-21 = 0.

( here if it is 1 then leave row and column of 1.
1 : if you remove u get 5 6
2. 1
then multiply and subtract.)

* you should always keep + - + order.
it means 1(2(3)-4(5)) - 3((2)-4(5)) +4((2)-5(2))
+ - +


now for adj u have to remember one order it is

+ - +
- + -
+ - +

eg: if 1 2 3
4 5 6
3 2 1
follow these steps :

1. first u need to do det.
eg: for 1 det is 5(6)-6(2) = 30-12 = 8. (remove row and column of 1)

2. like this find det of all numbers in matrix.
if 1det is 8
2det is 4 ,3det is 6, 4det is 12 ,5. Dt is 3, 6dt 5, ..........

3.write det answer in place of number

8 4 6 ( 1 dt is 8 so keep 8 in 1 place )
12 3 5
_ _ _


4. keep it in + - + order.
- + -
+ - +
eg: +(8) -(4) +(6)
-(12) +(3) -(5)
_ _ _

then its adj is 8 -4 6
-12 3 5
_ _ _.