To determine the correct option, let's analyze the given matrix
A :
A=[] 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
A1,3 and
A3,1 ).
So, Option A is incorrect.
Option B: A is a skew-symmetric matrix
A matrix is skew-symmetric if it satisfies the condition
AT=−A, where
AT denotes the transpose of matrix A.
Let's compute the transpose of
A :
−A=[] We see that
AT≠−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
A2=I, where
I is the identity matrix.
We compute
A2 as follows:
Thus,
A2=I.
Therefore, Option D is correct.
In conclusion, the correct answer is:
Option D:
A2=I