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: Think about what multiplying a matrix by 2 does to each row.
When we form $2A$, every single entry of $A$ gets doubled. That means every row of the matrix is scaled by $2$, one row at a time.

Step 2: Recall how row scaling affects the determinant.
A basic rule of determinants says that scaling one row of a matrix by a constant $c$ scales the whole determinant by that same $c$. If we scale a second row by $c$ as well, the determinant is multiplied by $c$ again, and so on for every row that gets scaled.

Step 3: Apply this rule four times, once for each row.
Matrix $A$ has $4$ rows, and each one is being scaled by $2$ when we form $2A$. So we multiply the determinant by $2$ a total of four times, once per row:
\[ \det(2A) = 2 \times 2 \times 2 \times 2 \times \det(A) = 2^4 \det(A) \]

Step 4: Plug in the given determinant.
We are told $\det(A) = 3$. So
\[ \det(2A) = 16 \times 3 \]

Step 5: Work out the multiplication.
\[ 16 \times 3 = 48 \]

Step 6: State the result.
The determinant of $2A$ works out to $48$, since scaling every row of a $4 \times 4$ matrix by $2$ scales its determinant by $2^4=16$.
\[ \boxed{48} \]