If `A` and `B` are matrices of order 3 and `|A|=5,|B|=3`, the `|3AB|`

Explanation

To find the determinant of the matrix \( |3AB| \), we will use the properties of determinants. Here’s the step-by-step solution:

### 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: Apply the scalar multiplication property
For any scalar \( k \) and a matrix \( A \) of order \( n \),
\[ |kA| = k^n |A| \]
where \( n \) is the order of the matrix. In this case, since \( A \) and \( B \) are both 3x3 matrices, \( n = 3 \).

### Step 3: Calculate \( |3AB| \)
Using the properties mentioned:
\[ |3AB| = |3I \cdot AB| = |3I| \cdot |AB| \]
where \( I \) is the identity matrix of order 3.

### Step 4: Calculate \( |3I| \)
Since \( |3I| = 3^3 = 27 \) (because the determinant of a scalar multiple of the identity matrix is the scalar raised to the power of the order of the matrix),
\[ |3I| = 27 \]

### Step 5: Calculate \( |AB| \)
Using the property of determinants for the product of matrices:
\[ |AB| = |A| \cdot |B| = 5 \cdot 3 = 15 \]

### Step 6: Combine the results
Now we can substitute back into our equation:
\[ |3AB| = |3I| \cdot |AB| = 27 \cdot 15 \]

### Step 7: Calculate the final determinant
Now we calculate:
\[ 27 \cdot 15 = 405 \]

Thus, the final result is:
\[ |3AB| = 405 \]

### Final Answer
\[ |3AB| = 405 \]

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