What Is the Determinant of a Scalar Times a Matrix?

If you multiply every entry of a square matrix by the same scalar, the determinant does not usually get multiplied by that scalar just once. For an $n\times n$ matrix, the scalar contributes once for each of the $n$ rows (equivalently, each of the $n$ columns).

The rule is

$$\det(cA)=c^n\det(A),$$

where $A$ is an $n\times n$ matrix and $c$ is a scalar. The exponent is the matrix size. This differs from multiplying a single row by $c$, which multiplies the determinant by just $c$.

Why the scalar is raised to the matrix size

Write the columns of $A$ as $a_1,a_2,\ldots,a_n$. The columns of $cA$ are then $ca_1,ca_2,\ldots,ca_n$. The determinant is linear in each column when the other columns are held fixed. Pulling out the factor $c$ from each column gives

$$\det(cA)=\det(ca_1,ca_2,\ldots,ca_n) =c^n\det(a_1,a_2,\ldots,a_n) =c^n\det(A).$$

There are $n$ factors of $c$, one from each column. The same reasoning works with rows. This argument also applies when $c=0$: the zero matrix has determinant $0$ for any positive matrix size.

Worked example: scaling a 2 by 2 matrix

Take

$$A=\begin{pmatrix}1&2\\3&4\end{pmatrix}, \qquad c=3.$$

First find the determinant of $A$:

$$\det(A)=1\cdot4-2\cdot3=-2.$$

Since $A$ is a $2\times2$ matrix, the scaling rule gives

$$\det(3A)=3^2\det(A)=9(-2)=-18.$$

You can check by calculating the determinant directly. The scaled matrix is

$$3A=\begin{pmatrix}3&6\\9&12\end{pmatrix},$$

so

$$\det(3A)=3\cdot12-6\cdot9=36-54=-18.$$

The result would be different if you used $3\det(A)=-6$; that would account for scaling just one row or column, not every entry of the matrix.

Related scaling rules

The determinant rules depend on exactly what is being multiplied:

Operation	Effect on the determinant
Multiply the entire $n\times n$ matrix by $c$	$\det(cA)=c^n\det(A)$
Multiply one row by $c$	The determinant is multiplied by $c$
Multiply one column by $c$	The determinant is multiplied by $c$
Multiply square matrices $A$ and $B$	$\det(AB)=\det(A)\det(B)$

For example, scaling one row of a $3\times3$ matrix by $c$ contributes one factor of $c$, whereas scaling the entire matrix contributes $c^3$. The product rule is also distinct: if $A$ and $B$ are square matrices of the same size, the determinant of their matrix product is the product of their determinants.

Common mistakes

• Using $c\det(A)$ for the whole matrix. That formula misses the other $n-1$ factors. Use $c^n\det(A)$ when every entry is scaled.
• Using the number of entries as the exponent. For an $n\times n$ matrix, the exponent is $n$, not $n^2$.
• Confusing scalar multiplication with matrix multiplication. A scalar $c$ gives $cA$. If the other factor is itself a matrix, use the product rule when the matrices are square and the product is defined.
• Taking the determinant of a rectangular matrix. The standard determinant is defined only for square matrices, so the scaling formula for determinants does not apply to a rectangular matrix.

When a problem says “multiply a matrix by a constant,” the usual meaning is that the same scalar multiplies every entry. Under that interpretation, identify the matrix size and raise the scalar to that size.