If A and B are square matrices of order 3 such that |A| = - 1 and |B| = 3 , then the value of |3AB| is a) 27 b) -27 c) -81 d) 81

Options

A. 27
B. `-27`
C. `-81`
D. 81

Answer

To find the value of \(|3AB|\), we can use the properties of determinants. Let's break it down step by step.

### Step 1: Understand the properties of determinants
The determinant of a product of matrices is equal to the product of their determinants. That is:
\[ |AB| = |A| \cdot |B| \]

### Step 2: Use the property of scalar multiplication
When a matrix \(A\) is multiplied by a scalar \(k\), the determinant of the resulting matrix is given by:
\[ |kA| = k^n |A| \]
where \(n\) is the order of the matrix. In this case, since \(A\) and \(B\) are \(3 \times 3\) matrices, \(n = 3\).

### Step 3: Calculate \(|3AB|\)
Using the properties mentioned, we can express \(|3AB|\) as:
\[ |3AB| = |3I \cdot AB| = |3I| \cdot |AB| \]
where \(I\) is the identity matrix.

Now, applying the scalar multiplication property:
\[ |3I| = 3^3 = 27 \]

### Step 4: Calculate \(|AB|\)
Using the property of the determinant of a product:
\[ |AB| = |A| \cdot |B| \]
Given \(|A| = -1\) and \(|B| = 3\):
\[ |AB| = (-1) \cdot 3 = -3 \]

### Step 5: Combine the results
Now substituting back into our equation for \(|3AB|\):
\[ |3AB| = |3I| \cdot |AB| = 27 \cdot (-3) = -81 \]

### Final Answer
Thus, the value of \(|3AB|\) is:
\[ \boxed{-81} \]