MATH 240

Mon. October 7th, 2019

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Applications of the Determinant

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Cramer’s Rule

Given an $n\times n$ matrix $A$, by $A_{i}(b)$ we denote the matrix obtained by replacing the $i$th column of $A$ with $b$.

Cramer’s Rule: Let $A$ be an $n\times n$ invertible matrix. For any $b$ in $\R^n$, the solution to $Ax=b$ is given by $x_i=\dfrac{\text{det}A_i(b)}{\text{det}A}.$

This is not very useful for calculations, but makes some theoretical proofs simpler.

Proof. Consider $I_i(x)$.
$\begin{aligned} I_i(x)&=\begin{pmatrix} 1&0&...&x_1&...&0\\ 0&1&...&x_2&...&0\\ ...\\ 0&0&...&x_i&...&0\\ ...\\ 0&0&...&x_n&...&1\\ \end{pmatrix}\\ \\ \text{det}I_i(x)&=\begin{vmatrix} 1&0&...&x_1&...&0\\ 0&1&...&x_2&...&0\\ ...\\ 0&0&...&x_i&...&0\\ ...\\ 0&0&...&x_n&...&1\\ \end{vmatrix}\\ &=x_i\begin{vmatrix} 1&0&...&x_1&...&0\\ 0&1&...&x_2&...&0\\ ...\\ 0&0&...&1&...&0\\ ...\\ 0&0&...&x_n&...&1\\ \end{vmatrix}\\ &=x_i\begin{vmatrix} 1&0&...&0&...&0\\ 0&1&...&0&...&0\\ ...\\ 0&0&...&1&...&0\\ ...\\ 0&0&...&0&...&1\\ \end{vmatrix}\\ &=x_i\ \text{det}I_n\\ &=x_i\\ \\ AI_i(x)&=A\begin{bmatrix} e_1&e_2&...&x&e_{i+1}&...&e_n \end{bmatrix}\\ &=\begin{bmatrix} Ae_1&Ae_2&...&Ax&Ae_{i+1}&...&Ae_n \end{bmatrix}\\ &=\begin{bmatrix} Ae_1&Ae_2&...&b&Ae_{i+1}&...&Ae_n \end{bmatrix}\\ &=A_i(b)\\ \\ \text{det}A_i(b)&=\text{det}AI_i(x)\\ &=\text{det}A\ \text{det}I_i(x)\\ &=x_i\text{det}A\\ \\ x_i&=\dfrac{\text{det}A_i(b)}{\text{det}A}. \end{aligned}$

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Ex. Let $s$ be a parameter
$\begin{cases} 3sx_1-2x_2=4\\-6x_1+sx_2=1 \end{cases}$
and assume that the system has a unique solution.
$\begin{aligned} A&=\begin{pmatrix} 3s&-2\\-6&s \end{pmatrix}\\ \text{det}A&=3s^2-12\\ &=3(s^2-4)\\ &=3(s-2)(s+2)\\ s&\neq 2,-2\\ \\ A_1(b)&=\begin{pmatrix} 4&-2\\1&s \end{pmatrix}\\ \text{det}A_1(b)&=4s+2\\ \\ A_2(b)&=\begin{pmatrix} 3s&4\\-6&1 \end{pmatrix}\\ \text{det}A_2(b)&=3s+24\\ \end{aligned}$

By Cramer’s Rule:
$\begin{aligned} x_1&=\dfrac{4s+2}{3(s-2)(s+2)}\\ x_2&=\dfrac{3s+24}{3(s-2)(s+2)} \end{aligned}$

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Cramer’s Rule also provides a formula to find the inverse of matrices. (You really shouldn’t use this for computations because it’s harder than other methods we’ve learned.)
$A^{-1}=\dfrac{1}{\text{det}A}\begin{pmatrix} C_11&C_21&...&C_n1\\ C_12&C_22&...&C_n2\\ ...\\ C_1n&C_2n&...&Cnn \end{pmatrix}$
where $C_ij$ are cofactors that were defined previously. (The matrix on the right here is called the classical adjoint of $A$.)

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Geometric Applications

If $A$ is a $2\times 2$ matrix, the area of the parallelogram determined by the columns of $A$ is $\mid\text{det}A\mid$. This also applies to the volume of parallelopipeds for $3\times 3$ matrices, and for higher-dimensional equivalents of parallelograms and volume for larger square matrices.

By determined by the columns of $A$, we mean the parallelogram that has $(0,0)$ and each column of $A$ as vertices. The remaning vertices can be determined from this information.

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Let $T: \R^2\rightarrow\R^2$ be a linear transformation determined by the matrix $A$. If $S$ is an object in $\R^2$ with finite area, then the area of $T(S)$ is the product of $\mid\text{det}A\mid$ and the area of $S$. ($T(S)$ means applying $T$ to every point in $S$.)