WBJEE 2024 Math Paper

To determine the correct option, let's analyze the given matrix $A$ :$A=\begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$ First, let's check if $A$ is a null matrix.A null matrix is a matrix in which all entries are zero. Clearly, the given matrix $A$ is not a null matrix because it contains nonzero elements (e.g. -1 in positions $A_{1,3}$ and $A_{3,1}$ ).So, Option A is incorrect. Option B: A is a skew-symmetric matrixA matrix is skew-symmetric if it satisfies the condition $A^T=-A$, where $A^T$ denotes the transpose of matrix A.Let's compute the transpose of $A$ : $-A=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ We see that $A^T \neq -A$, hence $A$ is not a skew-symmetric matrix.Therefore, Option B is incorrect.Next, let's check if $A^{-1}$ exists.To check the invertibility of a matrix, we can compute its determinant. If the determinant is non-zero, the matrix is invertible; otherwise, it is not invertible. We compute the determinant of $A$ as follows: Since the determinant is non-zero, matrix $A$ is invertible, and $A^{-1}$ exists.So, Option C is incorrect.Finally, let's check if $A^2=I$, where $I$ is the identity matrix.We compute $A^2$ as follows: Thus, $A^2=I$.Therefore, Option D is correct.In conclusion, the correct answer is:Option D: $A^2=I$