Question

The determinant of a 4 × 4 matrix A is 3. The value of the determinant of 2A is
____________. (answer in integer)

Hint:

Use the property \(\det(kA) = k^n \det(A)\) for an \(n \times n\) matrix, where \(n = 4\) and \(k = 2\).

Answer 1

Correct Answer: 48

We are given that matrix \(A\) is a \(4 \times 4\) matrix with \(\det(A) = 3\). We need to find \(\det(2A)\).

Step 1: Recall the scalar multiplication property of determinants. For an \(n \times n\) matrix \(A\) and a scalar \(k\), scaling the matrix by \(k\) means every row of the matrix gets multiplied by \(k\). Since the determinant is linear in each row separately, multiplying all \(n\) rows by \(k\) multiplies the determinant by \(k^n\). So:

\[\det(kA) = k^n \det(A)\]

Step 2: Here \(n = 4\) (since \(A\) is \(4 \times 4\)) and \(k = 2\). Substituting these values:

\[\det(2A) = 2^4 \det(A)\]

Step 3: We know \(\det(A) = 3\), so:

\[\det(2A) = 2^4 \times 3 = 16 \times 3 = 48\]

Final Answer:

\[\boxed{\det(2A) = 48}\]