1. Key Concepts and Formulas

• Zero Matrix ($O$): A matrix where all its elements are zero.
• Identity Matrix ($I$): A square matrix with ones on the main diagonal and zeros elsewhere. For a $3 \times 3$ matrix, $I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$.
• Scalar Multiplication: To multiply a matrix by a scalar, multiply every element of the matrix by that scalar.
• Matrix Multiplication: For two matrices $A$ (size $m \times n$) and $B$ (size $n \times p$), their product $C = AB$ is an $m \times p$ matrix where each element $C_{ij}$ is the dot product of the $i$-th row of $A$ and the $j$-th column of $B$.
• Determinant of a Matrix ($|A|$ or $\det(A)$): A scalar value that can be computed for a square matrix. For a $3 \times 3$ matrix $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$, its determinant can be found by cofactor expansion, e.g., along the first row: $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$.
• Existence of Inverse ($A^{-1}$): A square matrix $A$ has an inverse $A^{-1}$ if and only if its determinant $|A|$ is non-zero. Such a matrix is called non-singular.

2. Step-by-Step Solution

The given matrix is: $A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$ We will systematically evaluate each option.

Step 1: Analyze Option (D) - $A$ is a zero matrix.

• What we are doing: Comparing the given matrix $A$ with the definition of a $3 \times 3$ zero matrix.
• Why we are doing this: To check if $A$ fits the definition of a zero matrix, which requires all its elements to be zero.
• Math: The $3 \times 3$ zero matrix is $O = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$. The given matrix is $A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$.
• Reasoning: For two matrices to be equal, all their corresponding elements must be equal. By inspection, elements like $A_{13} = -1$, $A_{22} = -1$, and $A_{31} = -1$ are non-zero. Therefore, $A$ is not a zero matrix.
• Conclusion: Option (D) is incorrect.

Step 2: Analyze Option (C) - $A^{-1}$ does not exist.

• What we are doing: Calculating the determinant of matrix $A$.
• Why we are doing this: The inverse of a matrix $A$ exists if and only if its determinant $|A|$ is non-zero. If $|A|=0$, the inverse does not exist.
• Math: We calculate the determinant of $A$ by expanding along the first row: $|A| = \left| \begin{matrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{matrix} \right|$ $|A| = 0 \cdot \left| \begin{matrix} -1 & 0 \\ 0 & 0 \end{matrix} \right| - 0 \cdot \left| \begin{matrix} 0 & 0 \\ -1 & 0 \end{matrix} \right| + (-1) \cdot \left| \begin{matrix} 0 & -1 \\ -1 & 0 \end{matrix} \right|$ $|A| = 0 - 0 + (-1) \cdot ((0)(0) - (-1)(-1))$ $|A| = (-1) \cdot (0 - 1)$ $|A| = (-1) \cdot (-1)$ $|A| = 1$
• Reasoning: The determinant of $A$ is $1$, which is a non-zero value. Therefore, the inverse $A^{-1}$ exists.
• Conclusion: Option (C) is incorrect.

Step 3: Analyze Option (B) - $A = (-1)I$.

• What we are doing: Calculating the matrix $(-1)I$ and comparing it with matrix $A$.
• Why we are doing this: To check if matrix $A$ is equal to the scalar multiple of the identity matrix by $-1$.
• Math: First, we calculate $(-1)I$: $(-1)I = (-1) \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$ Now, we compare this with $A$: $A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix} \quad \text{and} \quad (-1)I = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$
• Reasoning: For matrices to be equal, all corresponding elements must match. Comparing the elements, we see that $A_{11} = 0$ while $(-1)I_{11} = -1$, $A_{13} = -1$ while $(-1)I_{13} = 0$, etc. The matrices are clearly not equal.
• Conclusion: Option (B) is incorrect.

Step 4: Analyze Option (A) - $A^2 = I$.

• What we are doing: Performing matrix multiplication $A \times A$ to calculate $A^2$.
• Why we are doing this: This option is a direct statement about the result of $A$ multiplied by itself, so we must compute $A^2$ and compare it with the identity matrix $I$.
• Math: $A^2 = A \times A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$ Let $C = A^2$. We calculate each element $C_{ij}$ by taking the dot product of the $i$-th row of the first $A$ and the $j$-th column of the second $A$:
  • $C_{11} = (0)(0) + (0)(0) + (-1)(-1) = 0 + 0 + 1 = 1$
  • $C_{12} = (0)(0) + (0)(-1) + (-1)(0) = 0 + 0 + 0 = 0$
  • $C_{13} = (0)(-1) + (0)(0) + (-1)(0) = 0 + 0 + 0 = 0$
  • $C_{21} = (0)(0) + (-1)(0) + (0)(-1) = 0 + 0 + 0 = 0$
  • $C_{22} = (0)(0) + (-1)(-1) + (0)(0) = 0 + 1 + 0 = 1$
  • $C_{23} = (0)(-1) + (-1)(0) + (0)(0) = 0 + 0 + 0 = 0$
  • $C_{31} = (-1)(0) + (0)(0) + (0)(-1) = 0 + 0 + 0 = 0$
  • $C_{32} = (-1)(0) + (0)(-1) + (0)(0) = 0 + 0 + 0 = 0$
  • $C_{33} = (-1)(-1) + (0)(0) + (0)(0) = 1 + 0 + 0 = 1$ Thus, $A^2$ is: $A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
• Reasoning: The calculated matrix $A^2$ is precisely the $3 \times 3$ identity matrix $I$.
• Conclusion: Option (A) is correct.

3. Common Mistakes & Tips

• Careless Matrix Multiplication: This is a frequent source of errors. Always double-check each element calculation, especially signs and row-column pairings.
• Determinant Calculation Errors: Be meticulous with cofactor signs and arithmetic when expanding determinants. For matrices with many zeros, choose a row or column with the most zeros for expansion to simplify calculations.
• Misinterpreting Matrix Definitions: Ensure a clear understanding of what constitutes a zero matrix, an identity matrix, and scalar multiplication.
• Assuming Properties: Do not assume properties like invertibility or equality based on matrix appearance; always verify through calculation.

4. Summary

We systematically analyzed each given option to determine the correct statement about matrix $A$. We first confirmed that $A$ is not a zero matrix and that its inverse exists by calculating its non-zero determinant. We then verified that $A$ is not equal to $(-1)I$. Finally, by performing matrix multiplication $A^2 = A \times A$, we found that the result is the identity matrix $I$. This confirms that $A^2 = I$ is the only correct statement. A matrix for which $A^2 = I$ is known as an involutory matrix, implying it is its own inverse.

5. Final Answer

The final answer is $\boxed{A}$ which corresponds to option (A).