Question

If A is a skew symmetric matrx, then \(A^ {2021}\) is

• A. Row Matrix
• B. Symmetric Matrix
• C. Column Matrix
• D. Skew Symmetric Matrix

Answer 1

The Correct Option is D

If A is a skew symmetric matrix, it means that A is a square matrix such that \(A^T = -A\), where \(A^T\) is the transpose of matrix A.
Now, let's consider the power \(A^{2021}\)
Since A is skew symmetric, we can observe the pattern in the powers of A:
\(A^1 = A\)
\(A^2 = A \cdot A = A^T \cdot A = (-A) \cdot A = -A^2\)
\(A^3 = A \cdot A^2 = A \cdot (-A^2) = -(A \cdot A^2) = -A^3\)
From the pattern, we can deduce that \(A^k = (-1)^{k-1} \cdot A^k\), where k is an odd positive integer.
In the case of \(A^{2021}\), since 2021 is an odd number, we have:
\(A^{2021} = (-1)^{2021-1} \cdot A^{2021} = (-1)^{2020} \cdot A^{2021} = 1 \cdot A^{2021} = A^{2021}\)
This means that \(A^{2021}\) is equal to itself, which implies that \(A^{2021}\) is a skew symmetric matrix.
Therefore, the correct option is (D) Skew Symmetric Matrix.