Question

A= $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$which of the following is true?

• A. A is a periodic matrix with period 1
• B. A is an idempotent matrix
• C. A is a periodic matrix with period 2
• D. A is not an idempotent matrix

Answer 1

Answer: (A), (B)

A is an idempotent matrix

A=$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$is identity matrix and we know that $I=I^2=I^3$ and so on.

$I=I^2=I^{k+1}$

Now for $I^{k+1}=I$ we get k =0 which is not a positive integer.

Now to get the least positive integer we will equate $I^{k+1}$ to next higher power i.e. $I^2$

So, $I^{K+1}=I^2$

i.e., $k+1=2$

So $k=1$.

Since $K=1$ is the least integer satisfying the condition. So above matrix is a periodic matrix with period1. So option A is correct.

Also we know that for $A^{k+1}=A$, if $k=1$ satisfies the mentioned condition then the matrix is also called as idempotent matrix. So the above matrix is an idempotent matrix. So option (b) is also correct.