TIFR-2019-Maths-A: 10

Question

Let 

$$S=\left \{ x \in\mathbb{R} \mid x=Trace\:(A) \:for\:some\:A \in M_{4} (\mathbb{R}) such\:that\:A^{2}=A \right\}.$$

Then which of the following describes $S$?

• A. $S=\left \{ 0,2,4 \right \}$
• B. $S=\left \{ 0,1/2,1,3/2,2,5/2,3,7/2,4 \right \}$
• C. $S=\left \{ 0,1,2,3,4 \right \}$
• D. $S=\left \{ 0,4 \right \}$

Answer 1

$\text{Here, matrix A is  a real idempotent matrix of size $4 \times 4$ for which possible eigen values are 0 and 1. }$

$\text{Proof: Let, $v$ is an eigenvector associated with eigenvalue $\lambda.$}$

$\text{$\lambda v = Av$}$

$\text{$\lambda v = A^2v$}$

$\text{$\lambda v = \lambda^2v$}$

$\text{$  \lambda^2v – \lambda v = 0$}$

$\text{$  (\lambda^2- \lambda)v = 0$}$

$\text{$\lambda=0,1$}$

$\text{Here,}$

$\text{So, Trace(A) = $\lambda_1 + \lambda_2 + \lambda_3 + \lambda_4 $}$

$\text{Trace(A) = 4, when all $\lambda’s$ are 1, It is possible when A is identity(only possible non-singular matrix) }$

$\text{Trace(A) = 3, when any 3 $\lambda’s$ are 1 and other is 0 }$

$\text{Trace(A) = 2, when any 2 $\lambda’s$ are 1 and rest are 0 }$

$\text{Trace(A) = 1, when any 1 $\lambda’s$ is 1 and rest are 0 }$

$\text{Trace(A) = 0, when all  $\lambda’s$ are 0 }$

$\text{So, $S=\{0,1,2,3,4\}$}$