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MA1006/linear.tex
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243
MA1006/linear.tex
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\input{decls.tex}
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\title{Linear Algebra}
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\begin{document}
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\maketitle
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\section*{Allowable Operations on a Linear System}
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Solutions invariant.
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\begin{itemize}
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\item Multiply an equation by a non-zero scalar
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\item Swap two equations
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\item Add a multiple of one equation to another
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\end{itemize}
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\subsection*{Example}
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\begin{align*}
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&\systeme{
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x - 2y + 2z = 6,
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-x + 3y + 4z = 2,
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2x + y - 2z = -2
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}\\\\
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E_2 & \implies E_2 + E_1 \\
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E_3 & \implies E_3 + E_1 \\
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&\systeme{
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x - 2y + 2z = 6,
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y + 6z = 8,
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5y - 6z = -14
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}\\\\
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E_3 & \implies E_3 - 5E_2 \\
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&\systeme{
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x - 2y + 2z = 6,
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y + 6z = 8,
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z = \frac{3}{2}
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}\\\\
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E_1 & \implies E_1 - 2E_3 \\
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E_2 & \implies E_2 - 6E_3 \\
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&\systeme{
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x - 2y = 3,
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y = -1,
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z = \frac{3}{2}
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}\\\\
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E_1 & \implies E_1 + 2E_2 \\
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&\systeme{
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x = 1,
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y = -1,
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z = \frac{3}{2}
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}\\\\
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\end{align*}
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\section*{As Matrices}
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\begin{align*}
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\systeme{
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x + 2y = 1,
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2x - y = 3
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}
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\quad=\quad
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\begin{pmatrix}[cc|c]
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1 & 2 & 1 \\
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2 & -1 & 3
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\end{pmatrix}
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& \systeme{
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x - y + z = -2,
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2x + 3y + z = 7,
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x - 2y - z = -2
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} \quad=\quad \begin{pmatrix}[ccc|c]
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1 & -1 & 1 & -2 \\
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2 & 3 & 1 & 7 \\
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1 & -2 & -1 & -2
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\end{pmatrix} \\
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\grstep[R_3 - R_1]{R_2 - 2R_1} & \begin{pmatrix}[ccc|c]
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1 & -1 & 1 & -2 \\
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0 & 5 & -1 & 11 \\
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0 & -1 & -2 & 0
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\end{pmatrix} \\
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\grstep{5R_3 + R_2} & \begin{pmatrix}[ccc|c]
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1 & -1 & 1 & -2 \\
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0 & 5 & -1 & 11 \\
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0 & 0 & -11 & 11 \\
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\end{pmatrix} \\
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\grstep{-11^{-1}R_3} & \begin{pmatrix}[ccc|c]
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1 & -1 & 1 & -2 \\
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0 & 5 & -1 & 11 \\
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0 & 0 & 1 & -1
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\end{pmatrix} \\
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\grstep[R_1 - R_3]{R_2 + R_3} & \begin{pmatrix}[ccc|c]
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1 & -1 & 0 & -1 \\
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0 & 5 & 0 & 10 \\
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0 & 0 & 1 & -1
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\end{pmatrix} \\&
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\grstep{5^{-1}R_2} & \begin{pmatrix}[ccc|c]
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1 & -1 & 0 & -1 \\
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0 & 1 & 0 & 2 \\
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0 & 0 & 1 & -1 \\
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\end{pmatrix} \\
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\grstep{R_1 + R_2} & \begin{pmatrix}[ccc|c]
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1 & 0 & 0 & 1 \\
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0 & 1 & 0 & 2 \\
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0 & 0 & 1 & -1
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\end{pmatrix} \\
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= & \quad
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\left\{
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\subalign{
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x & ~= ~1 \\
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y & ~= ~2 \\
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z & ~= ~-1
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}
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\right.
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\end{align*}
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\section*{Row-Echelon Form}
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\begin{description}
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\item[Row-Echelon Form] The leading entry in each row is 1 and is further to the right than the previous row's leading entry,
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all 0 rows are at the end
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\item[Reduced Row-Echelon Form] every other entry in a column containing a leading 1 is 0
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\item[Theorem:] A matrix can be transformed to reduced row-echelon form using a finite number of allowable row operations
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\end{description}
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\subsection*{Example}
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\begin{align*}
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& \systeme{3x_1 + 2x_2 = 1,
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x_1 - x_2 = 4,
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2x_1 + x_2 = 5} = \begin{pmatrix}[cc|c]
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3 & 2 & 1 \\
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1 & -1 & 4 \\
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2 & 1 & 5
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\end{pmatrix} \\
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\grstep{R_1\swap R_2} & \begin{pmatrix}[cc|c]
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1 & -1 & 4 \\
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3 & 2 & 1 \\
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2 & 1 & 5
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\end{pmatrix} \\
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\grstep[R_2 - 3R_1]{R_3 - 2R_1} & \begin{pmatrix}[cc|c]
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1 & -1 & 4 \\
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0 & 5 & -11 \\
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0 & 3 & -3
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\end{pmatrix} \\
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\grstep{5^{-1}R_2} & \begin{pmatrix}[cc|c]
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1 & -1 & 4 \\
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0 & 1 & \frac{-11}{5} \\
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0 & 3 & -3
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\end{pmatrix} \\
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\grstep{R_3 - 2R_2} & \begin{pmatrix}[cc|c]
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1 & -1 & 4 \\
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0 & 1 & \frac{-11}{5} \\
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0 & 0 & \frac{18}{5}
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\end{pmatrix} \\
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= & \systeme{
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x_1 - x_2 = 4,
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x_2 = \frac{-11}{5},
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0x_1 + 0x_2 = \frac{18}{5}
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}
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\end{align*}
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\begin{align*}
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& \begin{pmatrix}[cccc|c]
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1 & -1 & 1 & 1 & 6 \\
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-1 & 1 & -2 & 1 & 3 \\
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2 & 0 & 1 & 4 & 1 \\
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\end{pmatrix} \\
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\grstep[R_2 + R_1]{R_3 - 2R_1} & \begin{pmatrix}[cccc|c]
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1 & -1 & 1 & 1 & 6 \\
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0 & 0 & -1 & 2 & 9 \\
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0 & 2 & -1 & 2 & -11
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\end{pmatrix} \\
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\grstep[R_2\swap R_3]{2^{-1}R_3} & \begin{pmatrix}[cccc|c]
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1 & -1 & 1 & 1 & 6 \\
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0 & 1 & \frac{1}{2} & 1 & \frac{-11}{2} \\
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0 & 0 & -1 & 2 & 9 \\
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\end{pmatrix} \\
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\grstep[R_1 + R_3]{R_2 - 2^{-1}R_3} & \begin{pmatrix}[cccc|c]
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1 & -1 & 0 & 3 & 15 \\
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0 & 1 & 0 & 0 & -10 \\
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0 & 0 & -1 & 2 & 9 \\
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\end{pmatrix} \\
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\grstep[-R_3]{R_1 + R_2} & \begin{pmatrix}[cccc|c]
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1 & 0 & 0 & 3 & 15 \\
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0 & 1 & 0 & 0 & -10 \\
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0 & 0 & 1 & -2 & -9 \\
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\end{pmatrix} \\
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= & \systeme{
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x_1 + 3x_4 = 5,
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x_2 = -10,
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x_3 - 2x_4 = -9
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} \\
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= & \left\{\substack{
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x_1 = 5 - 3t \\
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x_2 = -10 \\
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x_3 = -9 + 2t
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}\right.
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\end{align*}
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\section*{Determinants}
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The determinant of a matrix is defined only for square matrices.
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\[\det{A} \neq 0 \iff \exists \text{ a unique solution to the linear system represented by } A\]
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Let
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\[A = \begin{pmatrix}
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a_{11} & a_{12} & a_{1n} \\
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a_{21} & \ddots & \vdots \\
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a_{31} & \ldots & a_{3n} \\
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\end{pmatrix}
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\]
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\begin{description}
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\item[$i, j$ minor of $A$] an $n$x$n$ matrix constructed by removing the $i^\text{th}$ row and $j^\text{th}$ column of $A$ \\
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Denoted by $A_{ij}$
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\end{description}
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\begin{align*}
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& \det{A} \text{ where } n = 1. = a_{11} \\
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& \det{A} = a_{11}\det{A_{11}} - a_{12}\det{A_{12}} + ... + (-1)^{n+1}a_{1n} \tag{Laplace expansion of the first row} \\
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& \qquad \text{or laplace expansion along other row or column}
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\text{For } n = 2:& \\
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& \det{A} = a_{11}\cdot a_{22} - a_{12}\cdot a_{21}
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\end{align*}
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\begin{description}
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\item[Upper Triangular] lower left triangle is 0 - $d_{ij} = 0 \quad \forall{i > j}$
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\item[Lower Triangular] upper right triangle is 0 - $d_{ij} = 0 \quad \forall{i < j}$
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\item[Diagonal] only values on the diagonal - $d_{ij} = 0 \quad \forall{i \neq j}$ \\
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$\det{A} = \prod^{N}_{i=0}~a_{ij} \forall~\text{ row-echelon }A$
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\end{description}
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\begin{itemize}
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\item Multiplying a row of a square matrix $A$ by $r$ multiplies $\det{A}$ by $r$
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\item Swapping two rows of a square matrix $A$ multiplies $\det{A}$ by $-1$
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\item Adding a multiple of a row does not effect the determinant
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\end{itemize}
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\section*{Transposition}
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\begin{description}
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\item[$A^T$] $a^T_{ij} = a_{ji}~ \forall~i,j$
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\end{description}
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Note: $\det{A} = \det{A^T}~\forall~A$
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\section*{Matrix Multiplication}
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LHS has columns $=$ rows of RHS
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It's the cartesian product
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\[A\times B = (a_{i1}b_{j1} + a_{i2}b_{2j} + \ldots + a_{im}b_{mj})_{ij}\]
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\begin{align*}
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\begin{pmatrix}[c|c|c]
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2 & 1 + 1 & 3 + 6 \\
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4(2) & 4 + 1 & 3(4) + 6 \\
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0 & 2 & 2(6) \\
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\end{pmatrix} = \begin{pmatrix}
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2 & 2 & 9 \\
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8 & 5 & 18 \\
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0 & 2 & 12
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\end{pmatrix}
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\end{align*}
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\begin{align*}
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\begin{pmatrix}1 \\ 2 \\ 3 \end{pmatrix}\begin{pmatrix}1 & 2 & 3 & 4\end{pmatrix} + \begin{pmatrix}
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1 & 2 & 3 & 4 \\
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5 & 6 & 7 & 8 \\
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9 & 10 & 11 & 12 \\
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\end{pmatrix}
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\end{align*}
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\end{document}
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