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.

Answer 2

A matrix \( A \) is skew-symmetric if: \[ A^T = -A \] For any skew-symmetric matrix: - When raised to an odd power, the result is also a skew-symmetric matrix. - When raised to an even power, the result is a symmetric matrix. Given: \[ A^{2021} \] Since 2021 is an odd number, \( A^{2021} \) will also be a skew-symmetric matrix. Correct answer: Skew Symmetric Matrix 

Answer 3

A matrix A is skew-symmetric if A^T = -A.

We want to determine if A²⁰²¹ is symmetric or skew-symmetric.

Let's find the transpose of A²⁰²¹:

(A²⁰²¹)^T = (A * A * ... * A)^T (2021 times)

(A²⁰²¹)^T = A^T * A^T * ... * A^T (2021 times) (since (ABC)^T = C^TB^TA^T)

Since A is skew-symmetric, A^T = -A.

(A²⁰²¹)^T = (-A) * (-A) * ... * (-A) (2021 times)

(A²⁰²¹)^T = (-1)²⁰²¹ * (A * A * ... * A)

(A²⁰²¹)^T = -1 * A²⁰²¹

(A²⁰²¹)^T = -A²⁰²¹

Since the transpose of A²⁰²¹ is equal to -A²⁰²¹, A²⁰²¹ is a skew-symmetric matrix.

Answer:

Skew Symmetric Matrix