If A and B are square matrices of order 3 such that `|A|=-1,|B|=3`, the the determinant of 3 AB is equal to

Options

A. `-9`
B. `-27`
C. `-81`
D. `81`

Answer

To find the determinant of the matrix \(3AB\), where \(A\) and \(B\) are square matrices of order 3, we can follow these steps:

### Step 1: Use the property of determinants for the product of matrices
The determinant of the product of two matrices is equal to the product of their determinants. Therefore, we have:
\[ |AB| = |A| \cdot |B| \]

### Step 2: Substitute the known values
Given that \(|A| = -1\) and \(|B| = 3\), we can substitute these values into the equation:
\[ |AB| = (-1) \cdot 3 = -3 \]

### Step 3: Use the property of determinants for scalar multiplication
The determinant of a scalar multiplied by a matrix can be expressed as:
\[ |kA| = k^n |A| \]
where \(k\) is a scalar and \(n\) is the order of the matrix. In this case, our scalar \(k = 3\) and the order \(n = 3\).

### Step 4: Calculate the determinant of \(3AB\)
Now we can find the determinant of \(3AB\):
\[ |3AB| = 3^3 |AB| \]
Substituting \(|AB| = -3\):
\[ |3AB| = 27 \cdot (-3) \]

### Step 5: Compute the final value
Calculating the above expression gives:
\[ |3AB| = -81 \]

Thus, the determinant of \(3AB\) is \(-81\).

### Final Answer:
\[ \text{The determinant of } 3AB \text{ is } -81. \]

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