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GO Classes DPP | GATE CS, DA | Linear Algebra | Rank & Null Space

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Let $\mathrm{A}$ be an $\mathrm{n}$ $\times$ $\mathrm{n}$ real matrix for which two distinct non-zero $\mathrm{n}$-dimensional real column vectors $\mathrm{v}_1$ and $\mathrm{v}_2$ satisfy the relation $\mathrm{A}{\mathrm{v}_1} =\mathrm{Av}_2 $.

Which of the folllowing statement(s) about $\mathrm{A}$ is(are) true?

  1. At least one eigenvalue of $\mathrm{A}$ is zero.
     
  2. $\mathrm{A}$ is not full rank.
     
  3. Columns of $\mathrm{A}$ are linearly independent.
     
  4. The determinant of $\mathrm{A}$ is zero.

2 Answers

1 1 vote

Given that $Av_1=Av_2$,

where $v_1$ and $v_2$ are two distinct nonzero vectors.

Subtracting, we get

$Av_1-Av_2=0$

which gives

$A(v_1-v_2)=0$.

Since $v_1\neq v_2$, we have

$v_1-v_2\neq 0$.

Thus, the homogeneous system $Ax=0$ has a nontrivial solution.

Hence, the null space of $A$ contains a nonzero vector, so $A$ is not invertible.

Therefore, $A$ is not full rank.

So, statement B is true.

Also, for an $n\times n$ matrix, if $A$ is not invertible, then

$\det(A)=0$.

So, statement D is true.

Further, if $\det(A)=0$, then $0$ is an eigenvalue of $A$.

So, statement A is true.

Now if $A$ is not invertible, its columns cannot be linearly independent.

So, statement C is false.

Therefore, the correct statements are $\boxed{\text{A, B and D are true}}$.

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