Let A and B be (3 `xx` 3) matrices with det A = 4 and det B = 3.
What is det (2AB) equal to ?

Options

A. 96
B. 72
C. 48
D. 36

Explanation

To find the determinant of the matrix \(2AB\), we can use the properties of determinants. Here’s a step-by-step solution:

### Step 1: Use the property of determinants for scalar multiplication
When a matrix \(A\) is multiplied by a scalar \(k\), the determinant of the new matrix is given by:
\[ \text{det}(kA) = k^n \cdot \text{det}(A) \]
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 the matrix \(2AB\)
We can express the determinant of \(2AB\) as:
\[ \text{det}(2AB) = \text{det}(2 \cdot (AB)) \]

### Step 3: Use the property of determinants for the product of matrices
The determinant of a product of matrices is the product of their determinants:
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \]
Thus, we can write:
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) = 4 \cdot 3 = 12 \]

### Step 4: Combine the results
Now substituting back into the equation for \(\text{det}(2AB)\):
\[ \text{det}(2AB) = 2^3 \cdot \text{det}(AB) = 8 \cdot 12 = 96 \]

### Final Answer
Thus, the determinant of \(2AB\) is:
\[ \text{det}(2AB) = 96 \]

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