MATH 251 Lecture 10
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How to calculate determinants
Basics
- Go across the top row (left to right)
- Cross off that item's row and column, then calculate the determinant of the resulting 2×2 Matrix
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \begin{vmatrix}1&3&-1\\4&2&-2\\0&1&1\end{vmatrix} &= 1(2--2)-3(4-0)+-1(4-0) \\ &= 4-12-4 \\ &= -12 \end{align}}
Trick 1
You can go across any row OR column!
Be sure to follow the sign rules:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \begin{vmatrix}3&157&87\\1&15&73\\0&0&1\end{vmatrix} &= 0-0+1(45-157) \\ &= -112 \end{align}}
Trick 2
Factor stuff from any row or column
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \begin{vmatrix}x^3-3x^2 & x^3 & x^2+x^4 \\ x-3 & 1 & 0 \\ 3-x & 2 & x^2+1 \end{vmatrix} &= x^2(x-3)(x^2+1) \begin{vmatrix}1&x&1\\1&1&0\\-1&2&1\end{vmatrix} \\ &= x^2(x-3)(x^2+1) \left[ -1(x-2)+1(1--1)+0 \right] \\ &= x^2(x-3)(x^2+1)(4-x) \end{align}}
Trick 3
Flipping any two rows or columns only flips the sign
... not very useful though
Trick 4
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \begin{vmatrix}3y&2-xy&1+3y\\1&-2\end{vmatrix} &= \begin{vmatrix}0&2&1\\1&-2&1\\2&1&5\end{vmatrix} \\ &= -2(5-2)+1(1--4) \\ &= -6+5 \\ &= -1 \end{align}}
Notice in the first matrix that the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} terms are just multiples of the second row added to the constants in the second matrix.
Looking at the geometry, doing this only shifts the vectors over, keeping the same base and height values.
Transformation Lattices
Drawing lattice lines w/r/t vector in "cartesian style"
Transform them into x-y lattice by holding one variable constant, then plotting the resulting shape. Hold the other variable constant, and plot those too.
Derivative
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases}x = r\cos{\theta} \\ y = r\sin{\theta}\end{cases}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{d} x &= \frac{\partial x}{\partial r} \, \mathrm{d} r + \frac{\partial x}{\partial \theta} \, \mathrm{d} \theta = \cos{\theta} \mathrm{d} r + -r \sin{\theta} \mathrm{d} \theta \\ \mathrm{d} y &= \frac{\partial y}{\partial r} \, \mathrm{d} r + \frac{\partial y}{\partial \theta} \, \mathrm{d} \theta = \sin{\theta} \mathrm{d} r + r \cos{\theta} \mathrm{d} \theta \end{align}}
So the derivative can be rewritten as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}\cos{\theta} & -r\sin{\theta} \\ \sin{\theta} & r\cos{\theta} \end{pmatrix}}
Jacobian
The determinant of the transformation matrix derivative (calculated above)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\cos^2{\theta} + r\sin^2{\theta} = r}
Multidimensional Limits
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \left( x,y \right) = \frac{x^{2} + y^{2}}{\sqrt{x^{2} + y^{2} + 49} - 7}}
Does Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{(x,y)\to(0,0)} f(x,y)} exist?
We approach (0,0) from all directions around it, and determine that they all converge to the same value.
All elementary functions are continuous at all points in their natural domain … In other words, if evaluating a limit at a point gives a real number, DO IT!!!
Trick: convert coordinates
let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2=r^2} (distance to the origin), therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(r) = \frac{r^2}{\sqrt{r^2+49}-7}} (L'Hospital's Rule)
Trick: approach along a plane
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{(x,y,z) \to (0,0,0)} \frac{xy+7yz^2+8xz^2}{x^2+y^2+z^4}}
Set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=0} to approach along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to0} \frac{0}{x^2} = 0}
Try to approach it from a different direction: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to0} = \frac{x^2+0+0}{2x^2} = \frac{1}{2}}
Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \ne \frac{1}{2}} , the limit does not exist.