Every square matrix can be expressed as the sum of:

Question

Accepted Answer

Explanation:

1. **Justification for Option 3: A symmetric and a skew-symmetric matrix**

Every square matrix $ A $ can indeed be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. This is a well-known result in linear algebra. The decomposition is given by:

$$
   A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)
   $$

Here, $ \frac{1}{2}(A + A^T) $ is a symmetric matrix because it equals its transpose:

$$
   \left(\frac{1}{2}(A + A^T)\right)^T = \frac{1}{2}(A^T + (A^T)^T) = \frac{1}{2}(A^T + A) = \frac{1}{2}(A + A^T)
   $$

Similarly, $ \frac{1}{2}(A - A^T) $ is a skew-symmetric matrix because its transpose is the negative of itself:

$$
   \left(\frac{1}{2}(A - A^T)\right)^T = \frac{1}{2}(A^T - (A^T)^T) = \frac{1}{2}(A^T - A) = -\frac{1}{2}(A - A^T)
   $$

Therefore, every square matrix can be decomposed into the sum of a symmetric and a skew-symmetric matrix, which confirms that Option 3 is correct.

2. **Explanation for why the other options are incorrect:**

- **Option 1: Two symmetric matrices**

A symmetric matrix is one that equals its transpose. While it is possible to express some matrices as the sum of two symmetric matrices, it is not a general rule that applies to all square matrices. The decomposition into symmetric and skew-symmetric matrices is more universally applicable.

- **Option 2: Two skew-symmetric matrices**

A skew-symmetric matrix is one that equals the negative of its transpose. It is not possible for every square matrix to be expressed as the sum of two skew-symmetric matrices. This is because the sum of two skew-symmetric matrices is also skew-symmetric, and not all square matrices are skew-symmetric.

- **Option 4: None of these**

This option is incorrect because, as demonstrated, every square matrix can indeed be expressed as the sum of a symmetric and a skew-symmetric matrix, making Option 3 the correct choice.

In conclusion, Option 3 is the correct answer because it accurately describes the decomposition of any square matrix into a symmetric and a skew-symmetric matrix, a fundamental concept in linear algebra.