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

Answer: (D)

Skew Symmetric Matrix

Answer 2

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.