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

An alternative way to see this result is to think about what happens to the determinant when we scale a matrix, rather than jumping straight to the formula.

Step 1: Think in terms of rows. The matrix \(2A\) is obtained from \(A\) by multiplying every single row of \(A\) by 2. Since \(A\) has 4 rows (it is \(4 \times 4\)), this means we perform the row-scaling operation 4 times, once for each row.

Step 2: Apply the effect of row scaling on determinants. A well-known property of determinants states that if you multiply just one row of a matrix by a constant $c$, the determinant of the resulting matrix gets multiplied by $c$ as well. So scaling one row by 2 doubles the determinant. Since all 4 rows are independently scaled by 2, the determinant gets multiplied by 2 a total of 4 times, giving a factor of $2^4 = 16$.

Step 3: Combine with the given determinant. We are told $\det(A) = 3$. Multiplying this by the scaling factor found above:

$\det(2A) = 16 \times \det(A) = 16 \times 3 = 48$

Conclusion: This confirms the result using the row-scaling logic rather than direct formula substitution. The determinant of $2A$ is $48$, matching the general rule $\det(kA) = k^n\det(A)$ for an $n \times n$ matrix.