### 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}

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