Below is a question from a competitive exam

Let $c_1,....c_n$ be scalars, not all zero, such that $\sum_{i=1}^{n}c_ia_i=0$ where $a_i$ are the column vectors in $R^n$.

Consider the set of linear equations


where $A=[a_1....a_n]$ and $b=\sum_{i=1}^{n}a_i$. The set of equations has

  1. a unique solution at $x=J_n$ where $J_n$ denotes an n-dimensional vector of all 1.
  2. No Solution
  3. Infinitely many solutions
  4. Finitely many solutions.

**My Attempt :* From given information, it is clear that the columns of the matrix A are dependent and hence there exists a non-zero vector $x_n$ where it denotes solution to the null space, such that $Ax_n=0$...(a)

Now, it is also given that $b=\sum_{i=1}^{n}a_i$, which means a column vector of all 1,say $x_p$ a particular solution to this b, is such that $Ax_p=b$ ..(b)

From (a) and (b),if c is a constant,then $A(x_p+c.x_n)=b$ and hence this system seems to have infinite solutions.

Is my attempt correct?


Yes, your attempt is correct.

There are infinitely many solutions. The matrix $A$ is singular and we have illustrated a particular solution, hence there must be infinitely many solutions.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.