If A is skew symmetric matrix of order 3 then the value of `|A|` is

Options

A. 3
B. 0
C. 9
D. 27

Answer

To solve the problem, we need to determine the value of the determinant of a skew-symmetric matrix of order 3. Let's go through the solution step by step:

### Step 1: Definition of Skew-Symmetric Matrix
A matrix \( A \) is called skew-symmetric if it satisfies the condition:
\[ A^T = -A \]
where \( A^T \) is the transpose of matrix \( A \).

### Step 2: Properties of Determinants
One important property of determinants is that the determinant of a transpose of a matrix is equal to the determinant of the original matrix:
\[ |A^T| = |A| \]

### Step 3: Applying the Skew-Symmetric Property
Since \( A \) is skew-symmetric, we can apply the property:
\[ |A^T| = |-A| \]
Using the property of determinants, we know that:
\[ |-A| = (-1)^n |A| \]
where \( n \) is the order of the matrix. In our case, since \( A \) is a 3x3 matrix (order 3), we have:
\[ |-A| = (-1)^3 |A| = -|A| \]

### Step 4: Setting Up the Equation
From the properties we have established, we can write:
\[ |A| = |-A| = -|A| \]

### Step 5: Solving the Equation
Now we have the equation:
\[ |A| = -|A| \]
Adding \( |A| \) to both sides gives:
\[ |A| + |A| = 0 \implies 2|A| = 0 \implies |A| = 0 \]

### Conclusion
Thus, the value of the determinant of a skew-symmetric matrix of order 3 is:
\[ \boxed{0} \]

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