Discussion
A 2×2 matrix \(Z = \begin{bmatrix} Z_{1} & Z_{2} \\ Z_{3} & Z_{4} \end{bmatrix}\) has the following properties:
(i) \(Z_{i} \in R\) for all \(i = 1, 2, 3, 4\);
(ii) \(Z_{i} \neq 0\) for some \(i = 1, 2, 3, 4\); and
(iii) \(Z = Z^{2} = Z^{3} = \cdots\)
Which one of the following options is CORRECT?
Answer: C
The correct option is \(Z = Z^{-1}\) for some \(Z\).
Property (iii) says \(Z^{2} = Z\): the matrix is idempotent, and (ii) says it is not the zero matrix. The identity matrix \(I\) satisfies all three properties, and \(I^{-1} = I\), so there is a \(Z\) equal to its own inverse.
That one example disposes of the first two options. \(I\) is invertible, so it is false that the inverse exists for no \(Z\). And \(I^{transpose} = I = I^{-1}\), so the transpose is not different from the inverse for every \(Z\).
The last option fails for \(Z = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\), which is non-zero and idempotent: with \(h = (0, 1)^{transpose}\), \(h^{transpose}Zh = 0\), which is not greater than 0.
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