Consider the matrix `A=[{:(0,-h,-g),(h,0,-f),(g,f, 0):}]` STATEMENT-1 : Det A = 0 STATEMENT-2 :The value of the determinant of a skew symmetric matrix of odd order is always zero.

Options

A. Statement - 1 is True, Statement - 2 is True' Statement - 2 is a correct explanation for Statement - 1
B. Statement - 1 is True, Statement - 2 is True, Statement - 2 is Not a correct explanation for Statement - 1
C. Statement - 1 is True, Statement - 2 is False
D. Statement - 1 is False, Statement - 2 is True

Answer

To solve the problem, we need to find the determinant of the given skew-symmetric matrix \( A \) and verify the statements provided.

The matrix \( A \) is given as:
\[ A = \begin{pmatrix} 0 & -h & -g \\ h & 0 & -f \\ g & f & 0 \end{pmatrix} \]

### Step 1: Calculate the Determinant of Matrix \( A \)

The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} 0 & -h & -g \\ h & 0 & -f \\ g & f & 0 \end{pmatrix} \]
We can identify:
- \( a = 0, b = -h, c = -g \)
- \( d = h, e = 0, f = -f \)
- \( g = g, h = f, i = 0 \)

Now, substituting these values into the determinant formula:
\[ \text{det}(A) = 0 \cdot (0 \cdot 0 - (-f) \cdot f) - (-h) \cdot (h \cdot 0 - (-f) \cdot g) + (-g) \cdot (h \cdot f - 0 \cdot g) \]

### Step 2: Simplify the Determinant Expression

Breaking it down:
1. The first term is \( 0 \cdot (0 + f^2) = 0 \).
2. The second term simplifies to:
\[ h \cdot (0 + fg) = hfg \]
3. The third term simplifies to:
\[ -g \cdot (hf) = -ghf \]

Putting it all together:
\[ \text{det}(A) = 0 + hfg - ghf = hfg - ghf = 0 \]

Thus, we conclude:
\[ \text{det}(A) = 0 \]

### Step 3: Verify the Statements

**Statement 1:** \( \text{det}(A) = 0 \) is correct.

**Statement 2:** The value of the determinant of a skew-symmetric matrix of odd order is always zero. Since our matrix \( A \) is skew-symmetric and of order 3 (which is odd), this statement is also correct.

### Conclusion

Both statements are correct, and the determinant of the skew-symmetric matrix \( A \) is indeed zero.