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

Explanation

To find the value of \( |3AB| \) given that \( |A| = 5 \) and \( |B| = 3 \), we can use properties of determinants. Let's break down the solution step by step.

### Step 1: 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 \( n \) is the order of the matrix. In this case, since \( A \) is a \( 3 \times 3 \) matrix, \( n = 3 \).

### Step 2: Apply the property to \( |3AB| \)
We can express \( |3AB| \) as:
\[ |3AB| = |3I \cdot AB| = |3I| \cdot |AB| \]
where \( I \) is the identity matrix. Using the property from Step 1, we have:
\[ |3I| = 3^3 = 27 \]
Thus,
\[ |3AB| = 27 |AB| \]

### Step 3: Use the property of determinants for the product of matrices
Next, we apply the property of determinants for the product of two matrices:
\[ |AB| = |A| \cdot |B| \]
Substituting the known values:
\[ |AB| = |A| \cdot |B| = 5 \cdot 3 = 15 \]

### Step 4: Substitute back to find \( |3AB| \)
Now, substituting \( |AB| \) back into the equation for \( |3AB| \):
\[ |3AB| = 27 \cdot |AB| = 27 \cdot 15 \]

### Step 5: Calculate the final value
Calculating \( 27 \cdot 15 \):
\[ 27 \cdot 15 = 405 \]

### Final Answer
Thus, the value of \( |3AB| \) is:
\[ |3AB| = 405 \]

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