Question

The determinant of a \(4 \times 4\) matrix \(A\) is 3. The value of the determinant of \(2A\) is ____________.

Hint:

For an \(n \times n\) matrix, \(\det(kA) = k^n \det(A)\).

Answer 1

Correct Answer: 48

Step 1: Recall the scalar multiplication property of determinants.
For an \(n \times n\) matrix \(A\) and a scalar \(k\), if we multiply every entry of \(A\) by \(k\), the determinant scales as
\[ \det(kA) = k^n \det(A) \]
This happens because every row of the matrix gets multiplied by \(k\) when we form \(kA\), and multiplying a single row of a matrix by \(k\) multiplies the whole determinant by \(k\). Since all \(n\) rows get scaled, the determinant picks up a factor of \(k\) a total of \(n\) times.

Step 2: Identify the values given in the question.
Here the matrix \(A\) is a \(4 \times 4\) matrix, so
\[ n = 4 \]
The scalar multiplying \(A\) is
\[ k = 2 \]
since we want the determinant of \(2A\). We are told that
\[ \det(A) = 3 \]

Step 3: Apply the formula from Step 1.
\[ \det(2A) = 2^4 \det(A) \]

Step 4: Substitute the known value and simplify.
\[ 2^4 = 16 \]
So,
\[ \det(2A) = 16 \times 3 = 48 \]

Step 5: Final Answer.
The determinant of \(2A\) is 48.
\[ \boxed{48} \]