This is a beautiful sample problem from ISI MStat PSB 2018 Problem 1. This is based on finding the real solution of a system of homogeneous equations. We provide a detailed solution with prerequisites mentioned explicitly.
Find all real solutions for the system of equations
We are given the system of homogeneous equations as follows ,
Let ,
As A is a matrix so ,if the Rank of A is <3 then it has infinitely many solution and
if Rank of A is 3 then it has only trivial solution i.e
Let's try to find the rank of matrix A from it's row echelon form ,
Now see when the determinant of A =0 as then the Rank(A) will be <3 and then it has infinitely many solutions
Det(A)=
So, if then Rank(A) <3 hence it has infinitely many solutions
Now from here we can say that if then Rank (A) =3 then the system of homogeneous equations has only trivial solution i.e
For system of homogeneous equation is as follows ,
Solving this we get and
. Hence solution space is {
} ,
.
Similarly , for we have solution space {
} and {
} respectively .
Therefore , real solutions (x1,x2,x3,λ) for the system of equations are ,
and
,
.
This is a beautiful sample problem from ISI MStat PSB 2018 Problem 1. This is based on finding the real solution of a system of homogeneous equations. We provide a detailed solution with prerequisites mentioned explicitly.
Find all real solutions for the system of equations
We are given the system of homogeneous equations as follows ,
Let ,
As A is a matrix so ,if the Rank of A is <3 then it has infinitely many solution and
if Rank of A is 3 then it has only trivial solution i.e
Let's try to find the rank of matrix A from it's row echelon form ,
Now see when the determinant of A =0 as then the Rank(A) will be <3 and then it has infinitely many solutions
Det(A)=
So, if then Rank(A) <3 hence it has infinitely many solutions
Now from here we can say that if then Rank (A) =3 then the system of homogeneous equations has only trivial solution i.e
For system of homogeneous equation is as follows ,
Solving this we get and
. Hence solution space is {
} ,
.
Similarly , for we have solution space {
} and {
} respectively .
Therefore , real solutions (x1,x2,x3,λ) for the system of equations are ,
and
,
.