Math Insight

Geometric properties of the determinant

The determinant of a square matrix is a single number that, among other things, can be related to the area or volume of a region. In particular, the determinant of a matrix reflects how the linear transformation associated with the matrix can scale or reflect objects. Here we sketch three properties of determinants that can be understood in this geometric context.

The effect of scaling a matrix

Since a linear transformation can always be written as $\vc{T}(\vc{x}) = A\vc{x}$ for some matrix $A$, applying a linear transformation to a vector $\vc{x}$ is the same thing as multiplying by a matrix. For square matrices $A$, the absolute value of the determinant captures how applying $\vc{T}$ expands or compresses objects. The nature of the expansion or compression depends on the underlying dimension. One-dimensional linear transformations expand length by a factor $| \det(A)|$, two-dimensional linear transformations expand area by a factor $| \det(A)|$, and three-dimensional linear transformations expand volume by a factor $| \det(A)|$.

How would the expansion of the linear transformation be affected if we scaled $A$? For example, if we doubled every entry of $A$, forming the matrix $B=2A$, how would the expansion of the linear transformation $\vc{\tilde{T}}=B\vc{x}$ compare with that of the original $\vc{T} = A\vc{x}$? In one dimension, the effect of doubling every vector would simply double the expansion of length by $\vc{\tilde{T}}$. But, in two dimensions, doubling both components of every vector would lead to increasing the area expansion of $\vc{\tilde{T}}$ by a factor of $2^2=4$, compared to the area expansion of $\vc{T}$. Similarly, doubling every component of the matrix associated with a three-dimensional $\vc{\tilde{T}}$ would expand volume $2^3=8$ times more than the original $\vc{T}$. Doubling every dimension of a three-dimensional object must lead to the eight-fold increase in volume.

Since this expansion is captured by $|\det(A)|$, we conclude that $|\det(2A)| = 2 |\det(A)|$ in one-dimension, $|\det(2A)| = 2^2 |\det(A)|$ in two-dimensions, and $|\det(2A)| = 2^3 |\det(A)|$ in three-dimensions. It turns out this works in higher dimensions (where a linear transformation expands higher-dimensional volume), and $|\det(2A)| = 2^n |\det(A)|$ in for an $n \times n$ matrix $A$.

There was, of course, nothing special about the number 2. By the same geometric argument, if we multiply every component of the matrix $A$ by a positive number $c$, we will change the resulting expansion of $\vc{T}$ by the factor $c$ in each of it dimensions. For an $n$-dimensional linear transformation, this multiplication will change its overall expansion by the factor $c^n$. Hence, we can conclude that for an $n \times n$ matrix $A$, $$|\det(cA)| = c^n |\det(A)|.$$

What about negative numbers and the sign of the determinant? The sign of the determinant determines whether a linear transformation preserves or reverses orientation. In one dimension, multiplying the one component of the matrix by a negative number would correspond to reflecting in that one dimension. Therefore, multiply by a negative number would change the size of the determinant. We can conclude that for one dimension, $\det(cA)=c\det(A)$ for any number $c$. (This result is painfully obvious since the determinant is just equal to the one number of $A$, but we discuss the one-dimensional case just to give intuition on higher dimensions.)

In two dimensions, the situation is different. If we multiplied every component of $A$ by a negative number, the change would correspond to flipping an object once across the $x$-axis and then once across the $y$-axis. Each of these reflections would change the orientation, but the two reflections cancel each other out. We cannot change the influence of a two-dimensional linear transformation $\vc{T}$ on orientation through multiplying all components of its matrix $A$ by a negative number. Combining this result with the expansion due to the magntiude of $c$, we conclude that $\det(cA) = c^2\det(A)$ in two dimensions. This equation summarizes how multiplication by a number cannot change the determinant sign in two dimensions.

In three-dimensions, multiplying the matrix $A$ by a negative number does change the influence of $\vc{T}$ on orientation. The net effect of a reflection across each of the three coordinate axes would still be a change of orientation. Since any odd number of reflections would change orientation, we see that multiplying the determinant by $c^n$ will capture the effect on orientation in $n$-dimensions. We conclude that for an $n \times $n matrix $A$ and a real number $c$, \begin{gather}\det(cA) = c^n \det(A).\end{gather}

The effect of multiplying matrices

The geometric interpretation allows us to quickly infer the determinant of a product $AB$ for $n\times n$ matrices $A$ and $B$. If we apply the linear transformation $\vc{S}(\vc{x}) = AB\vc{x}$ to an object, it's the same thing as first applying the linear transformation $\vc{\tilde{T}}(\vc{x}) = B\vc{x}$ and then applying the linear transformation $\vc{T}(\vc{x}) = A \vc{x}$. (This follows from the fact that matrix multiplication is associative.)

Since $\vc{T}$ expands by a factor of $|\det(A)|$ and $\vc{\tilde{T}}$ expands by a factor $|\det(B)|$, applying both transformations in succession must expand by a factor $|\det(A)||\det(B)|$. Moreover, the combination of the two transformations can reverse orientation only if just one of $\vc{T}$ or $\vc{\tilde{T}}$ orientation. We can summarize these observations with an equation for $\det(AB)$ that reflects the properties of the combined linear transformation $\vc{S}$: \begin{gather}\det(AB) = \det(A)\det(B).\end{gather}

The determinant of a matrix inverse

If one applies the linear transformation $\vc{T}(\vc{x})=A\vc{x}$ to an object, then applying the the linear transformation $\vc{T}^{-1}(\vc{x}) = A^{-1}\vc{x}$ should map everything back to the original object. Since $\vc{T}$ expands by the factor $|\det{A}|$, then $\vc{T}^{-1}$ must do the reverse, expanding by the factor $1/|\det(A)|$. The linear transformation $\vc{T}^{-1}$ must also reverse orientation just when $\vc{T}$ reverses orientation. In other words, the determinant of $A^{-1}$ must be related to the determinant of $A$ by \begin{gather}\det(A^{-1}) = \frac{1}{\det(A)}.\end{gather}