Question

A \(2\times2\) matrix \(Z=\begin{bmatrix}Z_1 & Z_2\\ Z_3 & Z_4\end{bmatrix}\) has the following properties:

• A. \(Z^{-1}\) does not exist for all \(Z\)
• B. \(Z^{transpose}\neq Z^{-1}\) for all \(Z\)
• C. \(Z=Z^{-1}\) for some \(Z\)
• D. For all \(Z\), \((h^{transpose}Zh)>0\) for all possible non-zero vector \(h\) with real elements

Hint:

A matrix satisfying \(Z^2=Z\) is called idempotent. The identity matrix is an idempotent matrix and is also its own inverse.

Answer 1

The Correct Option is C

Step 1: Understand the given condition.
Given that
\[ Z=Z^2=Z^3=\cdots \]
This means
\[ Z^2=Z \]
So, \(Z\) is an idempotent matrix.

Step 2: Check whether all such matrices are non-invertible.
An idempotent matrix may be non-invertible, for example
\[ Z=\begin{bmatrix}1&0\\0&0\end{bmatrix} \]
But it can also be invertible, for example
\[ Z=I=\begin{bmatrix}1&0\\0&1\end{bmatrix} \]
Thus, option (A) is not correct.

Step 3: Analyze the identity matrix case.
For
\[ Z=I \] we have
\[ Z^2=I^2=I=Z \]
So, the identity matrix satisfies the given condition.

Step 4: Check inverse of identity matrix.
For identity matrix,
\[ Z^{-1}=I^{-1}=I \]
Hence,
\[ Z=Z^{-1} \]

Step 5: Verify option (C).
Since there exists at least one matrix \(Z\), namely \(I\), for which
\[ Z=Z^{-1} \] option (C) is correct.

Step 6: Analyze option (D).
For every idempotent matrix, \(h^T Z h\) need not be positive for all non-zero real vectors \(h\).
For example, with
\[ Z=\begin{bmatrix}1&0\\0&0\end{bmatrix} \] and
\[ h=\begin{bmatrix}0\\1\end{bmatrix}, \] we get
\[ h^T Z h=0 \] which is not greater than \(0\).

Step 7: Final conclusion.
Therefore, the correct statement is
\[ \boxed{Z=Z^{-1}\text{ for some }Z} \]
Hence, the correct option is (C).