Am I misinterpreting this matrix determinant property?

Question

I was reading matrix determinant properties from wikipedia.

The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix.

However I am not able to realize it. What I find is $\det(cA) = c\det(A)$

For example, multiplying matrix by 2 and then taking the determinant of the resultant matrix:

$ 2\begin{bmatrix} 4 & 5 & 6 \\ 6 & 5 & 4 \\ 4 & 6 & 5 \\ \end{bmatrix}= \begin{bmatrix} 8 & 10 & 12 \\ 6 & 5 & 4 \\ 4 & 6 & 5 \end{bmatrix} $ and $ \begin{vmatrix} 8 & 10 & 12 \\ 6 & 5 & 4 \\ 4 & 6 & 5 \end{vmatrix}=60 $

Now first taking the determinant and then multiplying by 2 yields the same result:

$$ 2\begin{vmatrix} 4 & 5 & 6 \\ 6 & 5 & 4 \\ 4 & 6 & 5 \\ \end{vmatrix} = 2 \cdot 30 = 60 $$

Where I am mistaking?

Answer 1

Multiplying a matrix by a scalar $c$ amounts to multiplying each entry by $c$, not just the first row.

Answer 2

$$2 \begin{bmatrix} 4 & 5 & 6 \\ 6 & 5 & 4 \\ 4 & 6 & 5 \end{bmatrix} = \begin{bmatrix} 8 & 10 & 12 \\ 12 & 10 & 8 \\ 8 & 12 & 10 \end{bmatrix}.$$

You only multiplied the first row by 2.

Answer 3

You have $\det (cB) = \det (cI) \det B $ and you can see from the formula for $\det$ that $\det (cI) = c^n$.

Another way is to notice that $\det$ is a multilinear function of the columns (or rows), that is, we can write $\det(A) = f(a_1,...,a_n)$ where $a_k$ is the $k$th row of $A$, and $f$ is linear in each argument separately.

Then $\det(cA) = f(c a_1,...,c a_n) = c f(a_1, c a_2,...,c a_n) = c^2 f(a_1, a_2,...,c a_n)= \cdots = c^n f(a_1,...,a_n) = c^n \det A$.

Addendum:

You multiplied the matrix incorrectly.

$A=\begin{bmatrix} 4 & 5 & 6 \\ 6 & 5 & 4 \\ 4 & 6 & 5 \end{bmatrix}$, $2A = \begin{bmatrix} 8 & 10 & 12 \\ 12 & 10 & 8 \\ 8 & 12 & 10 \end{bmatrix}$. $\det A = 30, \det (2A) = 240$.

Answer 4

There are at least two ways to see that the answer is $c^n \det A$. One is by using the full expansion formula and noting that each term in the full expansion is a product of $n$ matrix entries and each matrix entry has multiplied by $c$. There is also a geometric explanation: The determinant gives the (signed) volume of the parallelopiped spanned by the columns of your matrix. If you dialate by a positive factor $c$, then the parallelopied is dialated by a factor of $c$ in each dimension, so the volume is dialated by a factor of $c^n$.