Determinants Of Scalar Multiples And Powers

Two Ways to Change a Matrix

A determinant records information about a square matrix, and familiar operations on the matrix can change that value in predictable ways. Two especially useful operations are multiplying every entry by the same scalar and taking repeated products of a matrix with itself. The resulting formulas are compact, but their exponents matter: scaling an \(n\times n\) matrix affects all \(n\) rows, while taking a power creates a product with repeated factors.

Throughout, let \(n\in\mathbb{N}\), let \(\mathbf{A}\in\mathbb{R}^{n\times n}\), and let \(c\in\mathbb{R}\). The scalar multiple \(c\mathbf{A}\) is the matrix whose entries are \(ca_{ij}\). Equivalently, each row of \(\mathbf{A}\) has been multiplied by \(c\). We use the multiplicative determinant law, established in the earlier material on determinants of matrix products: for square matrices \(\mathbf{A}\) and \(\mathbf{B}\) of the same size, \(\det(\mathbf{A}\mathbf{B})=\det(\mathbf{A})\det(\mathbf{B})\). We also use the row-scaling rule: multiplying one row by \(c\) multiplies the determinant by \(c\).

Proof. To prove the scalar formula, view \(c\mathbf{A}\) as the result of scaling each of the \(n\) rows of \(\mathbf{A}\) by \(c\). By the row-scaling rule, the first scaling contributes a factor \(c\), the second contributes another factor \(c\), and continuing through all \(n\) rows gives \(\det(c\mathbf{A})=\underbrace{c\cdots c}_{n\text{ factors}}\det(\mathbf{A})=c^n\det(\mathbf{A})\). This reasoning includes \(c=0\): scaling a row by zero makes that row zero and contributes a factor of zero.

For the power formula, first consider \(k=0\). By definition \(\mathbf{A}^0=\mathbf{I}_n\), and \(\det(\mathbf{I}_n)=1=\det(\mathbf{A})^0\). Now suppose the formula holds for a nonnegative integer \(k\). Since \(\mathbf{A}^{k+1}=\mathbf{A}^k\mathbf{A}\), the multiplicative determinant law gives \(\det(\mathbf{A}^{k+1})=\det(\mathbf{A}^k)\det(\mathbf{A})\). Using the induction hypothesis, this equals \(\det(\mathbf{A})^k\det(\mathbf{A})=\det(\mathbf{A})^{k+1}\). Thus the formula holds for \(k=0\), and whenever it holds for \(k\), it holds for \(k+1\). By induction it holds for every integer \(k\geq0\). \(\square\)

Why the Scalar Has Exponent \(n\)

The exponent in \(\det(c\mathbf{A})=c^n\det(\mathbf{A})\) counts the rows being scaled. It is not determined by the number of nonzero entries, the number of columns that happen to contain nonzero values, or the value of the determinant. Each of the \(n\) rows contributes one factor of \(c\), including when some rows contain zero entries already.

The formula also handles special scalar values without a separate exception. If \(c=1\), the matrix does not change and \(1^n=1\). If \(c=0\), the entire matrix becomes the zero matrix and the determinant is zero for \(n\geq1\). If \(c<0\), the sign of \(c^n\) depends on whether \(n\) is odd or even: an odd number of negative row factors gives a negative factor, while an even number gives a positive factor.

Determinants of Powers

The power formula is an immediate consequence of multiplying determinants. For example, \(\mathbf{A}^3=\mathbf{A}\mathbf{A}\mathbf{A}\), so its determinant is \(\det(\mathbf{A})\det(\mathbf{A})\det(\mathbf{A})\). The induction proof ensures that this reasoning works for every nonnegative integer power, not just the first few cases. No assumption that \(\mathbf{A}\) is invertible is required; the formula applies to every square matrix.

Combining Scaling and Powers

The two identities can be applied in sequence. For instance, if \(k\geq0\), then \(\det(c\mathbf{A}^k)=c^n\det(\mathbf{A}^k)=c^n\det(\mathbf{A})^k\). The exponent on \(c\) is still the size \(n\) of the matrix; the exponent on \(\det(\mathbf{A})\) is the power \(k\). Keeping those roles separate prevents a common mix-up.

Zero and Negative Powers

The convention \(\mathbf{A}^0=\mathbf{I}_n\) makes the power formula include \(k=0\): both sides equal \(1\). For negative powers, a condition is necessary because \(\mathbf{A}^{-1}\) is defined only when \(\mathbf{A}\) is invertible. The earlier result on determinants of inverses states that for an invertible square matrix, \(\det(\mathbf{A}^{-1})=1/\det(\mathbf{A})\). Consequently, if \(\mathbf{A}\) is invertible and \(m\in\mathbb{N}\), then

Here invertibility guarantees that \(\det(\mathbf{A})\ne0\), so the reciprocal is defined. The nonnegative-power formula itself has no such restriction: it remains valid even when \(\det(\mathbf{A})=0\).

Common Pitfall: Using the Wrong Exponent

A frequent error is to write \(\det(c\mathbf{A})=c\det(\mathbf{A})\), as if scaling the whole matrix scaled only one row. For the \(2\times2\) matrix in the first example, that incorrect rule would give \(3(-1)=-3\), while the actual determinant is \(-9\). The correction is to count all rows: for a \(2\times2\) matrix use \(3^2\), and, in general, for an \(n\times n\) matrix use \(c^n\).

A different mistake is to confuse the matrix size with the power. In \(\det(c\mathbf{A}^k)\), the correct result is \(c^n\det(\mathbf{A})^k\), not \(c^k\det(\mathbf{A})^n\). The scalar factor is applied to each of the \(n\) rows, while the matrix power repeats the determinant factor \(k\) times.