8. Let V be the set of all 2 x 2 matrices with the standard componentwise definitions for vector addition and scalar multiplication. Determine whether the following are or are not a vector space.
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![**Matrix Vector Spaces: Problem Review**
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### Problem 8
Consider \( V \) to be the set of all \( 2 \times 2 \) matrices with standard component-wise definitions for vector addition and scalar multiplication. Determine if the following sets are vector spaces.
**a) Set of Skew-Symmetric Matrices**
**Objective:**
Let \( V \) be the set of all skew-symmetric matrices, i.e., matrices that satisfy \( A^T = -A \).
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**Explanation:**
- **Vector Addition**: For any two skew-symmetric matrices \( A \) and \( B \), \( A + B \) should also be skew-symmetric.
- **Scalar Multiplication**: For any skew-symmetric matrix \( A \) and any scalar \( c \), the matrix \( cA \) should also be skew-symmetric.
### Detailed Analysis:
A matrix \( A \) is skew-symmetric if it satisfies the condition \( A^T = -A \). For example, a general \( 2 \times 2 \) skew-symmetric matrix can be represented as:
\[
A = \begin{pmatrix}
0 & a \\
-a & 0
\end{pmatrix}
\]
To confirm if the set \( V \) is a vector space, verify the following properties:
1. **Closure under addition**: If \( A \) and \( B \) are skew-symmetric matrices, then their sum \( A + B \) must also be skew-symmetric.
\[
A^T = -A \quad \text{and} \quad B^T = -B
\]
Therefore,
\[
(A + B)^T = A^T + B^T = -A + (-B) = -(A + B)
\]
This shows \( A + B \) is skew-symmetric.
2. **Closure under scalar multiplication**: If \( A \) is a skew-symmetric matrix and \( c \) is a scalar, then the product \( cA \) must be skew-symmetric.
\[
A^T = -A
\]
Then,
\[
(cA)^T = cA^T = c(-A) = -cA
\]
This shows \( cA \) is skew-symmetric.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d548be9-b70c-4345-ad21-fb156d23b920%2Ffb575314-97ad-45eb-9bcc-936cba54bdcd%2Fro9druu.png&w=3840&q=75)
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