Null Matrix

Q.

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

1. $55$

2. $58$

3. $56$

4. All of these

Q. If $A=\left[\begin{array}{ccc}1& w& w^2\\ w& w^2& 1\\ w^2& 1& w\end{array}\right]$ and $B=\left[\begin{array}{ccc}w& w^2& 1\\ w^2& 1& w\\ 1& w& w^2\end{array}\right]$ where $w$ is a complex cube root of unity, then " $AB$ and $BA$ both are null matrices." that statement is _____?

1. True
2. False

Q.

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

1. 57

2. 55

3. 58

4. 56

Q. If $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]$, then $A^2$=

1. Unit matrix
2. Null matrix
3. A
4. -A

Q.

Which of the following statements is correct about a null matrix?

1. It has more than one entries as zero

2. It has only one non-zero entry

3. It has all the entries as zero

4. It is represented as $O_{mxn}$ where mxn is the order of the matrix