Objectives: Topics
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Relationships between area, volume, determinants, and linear transformations.
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Cramer’s Rule for linear systems.
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Section 3.3 Applications of the Determinant
Handout 3.3 Applications of the Determinant
Objectives: Goals
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Use determinants to compute the area of a parallelogram or the volume of a parallelepiped, possibly under a given linear transformation.
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Use determinants and Cramer’s Rule to solve systems of linear equations.
In \(\mathbb{R}^2\text{,}\) determinants measure the (signed) area of the parallelogram formed by two vectors.
Illustration of a parallelogram in \(\mathbb{R}^2\) spanned by two vectors forming a \(2\times 2\) matrix whose determinant gives the signed area.
Area of the parallelogram \(\;=\;\det\!\begin{bmatrix} a \amp c\\ b \amp d \end{bmatrix}\;=\;ad-bc\text{.}\)
Theorem 3.17. Area from the Determinant.
The area of the parallelogram spanned by the columns of a \(2\times 2\) matrix \(A\) is \(\lvert \operatorname{det}(A)\rvert\text{.}\)
Theorem 3.18. Volume from the Determinant.
The volume of the parallelepiped spanned by the columns of an \(n\times n\) matrix \(A\) is \(\lvert \operatorname{det}(A)\rvert\text{.}\)
Example 3.19.
Calculate the area of the parallelogram determined by the points \((-2,3)\text{,}\) \((1,7)\text{,}\) \((6,-3)\text{,}\) and \((9,1)\text{.}\)
Example 3.20.
Calculate the area of the triangle determined by the points \((-3,-2)\text{,}\) \((0,3)\text{,}\) and \((4,-1)\text{.}\)
Theorem 3.21. Volume Scaling by a Linear Transformation.
If \(T_A:\mathbb{R}^n\to\mathbb{R}^n\) is the linear transformation defined by \(T_A(\vec x)=A\vec x\text{,}\) and \(S\) is any closed shape in \(\mathbb{R}^n\text{,}\) then
\begin{equation*}
\operatorname{volume}(T_A(S)) \;=\; \lvert \det(A)\rvert \cdot \operatorname{volume}(S).
\end{equation*}
Some observations:
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Recall from Section 1.9 that the matrix corresponding to a shearing linear transformation has the from \(\begin{bmatrix} 1 \amp k \\ 0 \amp 1 \end{bmatrix}\) or \(\begin{bmatrix} 1 \amp 0 \\ k \amp 1 \end{bmatrix}\text{.}\) Since each of these matrices has determinant 1, we see that shearing does not change the area of a shape.
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Note that \(S\) can be any closed shape. This includes parallelograms, polygons, circles, or crazy blobs, as long as the edge of the shape is connected.
Example 3.22.
Find the area of the interior \(E\) of the ellipse defined by the equation
\begin{equation*}
\left(\dfrac{2x - y}{2}\right)^2 + \left(\dfrac{y + 3x}{3}\right)^2 = 1\text{.}
\end{equation*}
Theorem 3.23. Cramer’s Rule.
Let \(A \) be an invertible \(n \times n\) matrix and let \(\vec b \in \mathbb{R}^n\text{.}\) Then the system \(A\vec x = \vec b\) has the unique solution \(\vec x\) whose entires are given by
\begin{equation*}
x_i = \dfrac{\det(A_i)}{\det(A)}\text{,}
\end{equation*}
where \(A_i\) is the matrix formed by replacing the \(i^\text{th}\) column of \(A\) by \(\vec b\text{.}\)
Example 3.24.
Use Cramer’s Rule to solve the system
\begin{equation*}
\begin{bmatrix} 1 \amp 3 \amp 2 \\ -2 \amp 1 \amp 4 \\ 0 \amp 4 \amp 3 \end{bmatrix}\vec x = \begin{bmatrix} 1 \\ 9 \\ 0 \end{bmatrix}\text{.}
\end{equation*}
