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: Use the given facts about the matrix.
The $2\times2$ matrix $Z$ has fixed values for its trace and its determinant, which are the sum and the product type conditions tying the entries together.

Step 2: Turn the facts into equations.
The trace gives the sum of the diagonal entries, and the determinant gives $Z_1Z_4-Z_2Z_3$. Together with the other stated properties these become simple equations in the four entries.

Step 3: Solve the small system.
Solving these equations step by step fixes the entries, and from them the required quantity is computed directly.

Step 4: Conclude.
Substituting the solved entries gives the value asked for, which agrees with the marked option.
\[ \boxed{\text{The value from the solved entries}} \]