Question

If $\omega$ is a complex cube root of unity with $\omega \ne 1$ and $P=\left[P_{ij}\right]$ is a $n\times n$ matrix with $P_{ij}={\omega }^{i+j}$. Then $P^2\ne 0$ when $n$ is equal to

• A. $55$
• B. $58$
• C. $56$
• D. All of these

Answer 1

The correct option is D

All of these

Explanation for the correct option

Step 1: Prerequisites for the solution

Given that $\omega$ is a complex cube root of unity then we know that ${\omega }^3=1\text{and}1+\omega +{\omega }^2=0$.

Since $P={\left[P_{ij}\right]}_{n\times n}$ and $P_{ij}={\omega }^{i+j}$ then we have,

When $n=1$,

$P={\left[P_{ij}\right]}_{1\times 1}$ and here $i=1\text{and}j=1$ then

$\begin{array}{rcl}{P}_{11}& =& {\omega }^{1+1}\\ & =& {\omega }^2\end{array}$

The $1\times 1$ matrix will have only one element which is $P_{11}$ then

$P=\left[{\omega }^2\right]$

Step 2: Simplification of the matrix when $\mathit{n}\mathbf{=}\mathbf{2}$

When $n=2$ then

Then the matrix will have four elements which are $P_{11},P_{12,}{P}_{21,}\text{and}{P}_{22}.$

Here, $P_{11}={\omega }^2,P_{12}={\omega }^3,P_{21}={\omega }^3,\text{and}{P}_{22}={\omega }^4$.

$\begin{array}{rcl}P& =& {\left[P_{ij}\right]}_{2\times 2}\\ & =& \left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]\\ & =& \left[\begin{array}{cc}{\omega }^2& {\omega }^3\\ {\omega }^3& {\omega }^4\end{array}\right]\\ & =& \left[\begin{array}{cc}{\omega }^2& {\omega }^3\\ {\omega }^3& {\omega }^3·\omega \end{array}\right]\end{array}$

Since ${\omega }^3=1$, we can write that

$P=\left[\begin{array}{cc}{\omega }^2& 1\\ 1& \omega \end{array}\right]$

Now, we need to check whether $P^2=0\text{or not}$

$\begin{array}{rcl}\therefore P^2& =& P\times P\\ & =& \left[\begin{array}{cc}{\omega }^2& 1\\ 1& \omega \end{array}\right]\left[\begin{array}{cc}{\omega }^2& 1\\ 1& \omega \end{array}\right]\\ & =& \left[\begin{array}{cc}{\omega }^4+1& {\omega }^2+\omega \\ {\omega }^2+\omega & 1+{\omega }^2\end{array}\right]\\ & =& \left[\begin{array}{cc}\omega +1& -1\\ -1& -\omega \end{array}\right]\end{array}$

From this, we know that $P^2\ne 0$.

Step 3: Simplification of the matrix when $\mathit{n}\mathbf{=}\mathbf{3}$

When $n=3$, then the matrix will have nine elements which are $P_{11},P_{12,}{P}_{13,}{P}_{21},P_{22},P_{23},P_{31},P_{32},\text{and}{P}_{33}$.

Here, $P_{11}={\omega }^2,P_{12}={\omega }^3,P_{13}={\omega }^4{,}_{}{P}_{21}={\omega }^3,P_{22}={\omega }^4,P_{23}={\omega }^5,P_{31}={\omega }^4,P_{32}={\omega }^5,\text{and}{P}_{33}={\omega }^6$ then

$\begin{array}{rcl}\therefore P& =& {\left[P_{ij}\right]}_{3\times 3}\\ & =& \left[\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right]\\ & =& \left[\begin{array}{ccc}{\omega }^2& {\omega }^3& {\omega }^4\\ {\omega }^3& {\omega }^4& {\omega }^5\\ {\omega }^4& {\omega }^5& {\omega }^6\end{array}\right]\\ & =& \left[\begin{array}{ccc}{\omega }^2& 1& \omega \\ 1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\end{array}\right]\end{array}$

Now, we need to check whether $P^2=0\text{or not}$

$\begin{array}{rcl}\therefore P^2& =& P\times P\\ & =& \left[\begin{array}{ccc}{\omega }^2& 1& \omega \\ 1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\end{array}\right]\left[\begin{array}{ccc}{\omega }^2& 1& \omega \\ 1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\end{array}\right]\\ & =& \left[\begin{array}{ccc}{\omega }^4+1+{\omega }^2& {\omega }^2+\omega +{\omega }^3& {\omega }^3+{\omega }^2+\omega \\ {\omega }^2+\omega +{\omega }^3& 1+{\omega }^2+{\omega }^4& \omega +{\omega }^3+{\omega }^2\\ {\omega }^3+{\omega }^2+\omega & \omega +{\omega }^3+{\omega }^2& {\omega }^2+{\omega }^4+1\end{array}\right]\\ & =& \left[\begin{array}{ccc}\omega +1+{\omega }^2& {\omega }^2+\omega +1& 1+{\omega }^2+\omega \\ {\omega }^2+\omega +1& 1+{\omega }^2+\omega & \omega +1+{\omega }^2\\ 1+{\omega }^2+\omega & \omega +1+{\omega }^2& {\omega }^2+\omega +1\end{array}\right]\\ & =& \left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\end{array}$

Here $P^2=0$

From this, we know that if a number $n$ is a multiple of $3$ then $P^2=0$ otherwise $P^2\ne 0$.

Hence, the correct option is (D).