Understanding Symmetric and Skew-Symmetric Matrices

Let's analyze the properties of the matrix (AB - BA) given that A and B are symmetric matrices of the same order. A matrix M is called symmetric if its transpose is equal to itself, i.e., $M^T = M$.

We are interested in the matrix C = AB - BA.

Transpose of (AB - BA)

To understand the nature of the matrix (AB - BA), let's find its transpose. The transpose of a sum or difference of matrices is the sum or difference of their transposes, i.e., $(X - Y)^T = X^T - Y^T$. Also, the transpose of a product of matrices is the product of their transposes in reverse order, i.e., $(XY)^T = Y^T X^T$.

So, we have:

$$ C^T = (AB - BA)^T $$ $$ C^T = (AB)^T - (BA)^T $$

Since A and B are symmetric matrices, we know that $A^T = A$ and $B^T = B$. Using the property of the transpose of a product:

$$ (AB)^T = B^T A^T = BA $$ $$ (BA)^T = A^T B^T = AB $$

Substituting these back into the expression for $C^T$:

$$ C^T = BA - AB $$

We can see that $BA - AB = -(AB - BA) = -C$.

Thus, we have $C^T = -C$. A matrix that satisfies this property is called a skew-symmetric matrix.

Properties of Skew-Symmetric Matrices

A skew-symmetric matrix has specific properties regarding its entries:

• The diagonal entries of a skew-symmetric matrix are always zero. Let C be a skew-symmetric matrix with entries $c_{ij}$. By definition, $c_{ij} = -c_{ji}$. For diagonal entries, $i = j$, so $c_{ii} = -c_{ii}$. This implies $2c_{ii} = 0$, which means $c_{ii} = 0$. Therefore, all diagonal entries are zero.
• The non-diagonal entries have the property $c_{ij} = -c_{ji}$ for $i \neq j$.

Evaluating the Statements

Now let's evaluate the given statements about the matrix (AB - BA), which we found to be skew-symmetric.

Statement 1: Its diagonal entries are equal but nonzero.

As established, the diagonal entries of a skew-symmetric matrix are always zero. While they are indeed equal (all equal to 0), they are zero, not nonzero. Therefore, Statement 1 is incorrect.

Statement 2: The sum of its non-diagonal entries is zero.

Consider the sum of all entries in the matrix C. The sum can be written as the sum of diagonal entries plus the sum of non-diagonal entries.

Sum of all entries = $\sum_{i=1}^n \sum_{j=1}^n c_{ij}$

Sum of diagonal entries = $\sum_{i=1}^n c_{ii} = \sum_{i=1}^n 0 = 0$ (since diagonal entries are zero).

Sum of non-diagonal entries = $\sum_{i \neq j} c_{ij}$

The sum of non-diagonal entries can be grouped into pairs $(i, j)$ and $(j, i)$ where $i \neq j$. For each pair, $c_{ij} + c_{ji}$. Since C is skew-symmetric, $c_{ij} = -c_{ji}$, so $c_{ij} + c_{ji} = c_{ij} + (-c_{ij}) = 0$.

The total sum of non-diagonal entries is the sum of these pairs: $\sum_{i < j} (c_{ij} + c_{ji}) = \sum_{i < j} 0 = 0$.

Therefore, the sum of the non-diagonal entries of the skew-symmetric matrix (AB - BA) is zero. Statement 2 is correct.

Conclusion

Based on the analysis, only Statement 2 is correct. Statement 1 is incorrect because the diagonal entries are zero.