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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

- Rank of a matrix
- System of homogeneous equation

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,Î») 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

- Rank of a matrix
- System of homogeneous equation

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,Î») for the system of equations are , and , .

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