Matrices

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• Matrices are a way of graphically storing certain data.

$\left[\begin{array}{c}x\\ y\end{array}\right]$

Matrix Addition

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]+\left[\begin{array}{cc}w& x\\ y& z\end{array}\right]=\left[\begin{array}{cc}w+a& x+b\\ y+c& z+d\end{array}\right]$

• 2 matrices can be added if and only if their dimensions are the same

Matrix Multiplication

• 2 matrices can be multiplied if and only if the number of columns on the first matrix is equal to the number of rows on the second matrix

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\times \left[\begin{array}{cc}w& x\\ y& z\end{array}\right]=\left[\begin{array}{cc}aw+by& ax+bz\\ cw+dy& cx+dz\end{array}\right]$

The Identity Matrix

• Basically the matrix equivalent of 1
• Any matrix times this is equal to the same matrix
• A matrix where everything is 0, but the main leading diagonal is “1”s $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

The Inverse Matrix

• The determinant of a matrix $X=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is as such: $\Delta =ad-bc$

• The inverse of a matrix $X$ is:

• $X{X}^{-1}$ is equal to the identity matrix $I$
• We can use this to determine an unknown matrix

Transformation Matrices

• These matrices are used to create linear transformations in a 2d plane
• They are placed on the left, e.g. $TX$
• The inverse is also placed on the left, e.g. $T^{-1}X$

Important Transformation Matrices

Reflection about x-axis

$\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

Reflection about y-axis

$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$

Reflection about the line $y=x$

$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Reflection about the line $y=-x$

$\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]$

Reflection about the line $y=[\tan \theta ]x+c$

$\left(\left[\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right]\left(X-\left[\begin{array}{c}0\\ c\end{array}\right]\right)\right)$

Rotation $\theta ˚$ anticlockwise about a point $x,y$

$\left(\left[\begin{array}{cc}\cos (\theta )& -\sin (\theta )\\ \sin (\theta )& \cos (\theta )\end{array}\right]\left(X-\left[\begin{array}{c}x\\ y\end{array}\right]\right)\right)+\left[\begin{array}{c}x\\ y\end{array}\right]$

Translation $a$ along x-axis and $b$ along y-axis

$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}a\\ b\end{array}\right]$

Rotation $\theta ˚$ clockwise about a point $x,y$

$\left(\left[\begin{array}{cc}\cos (\theta )& \sin (\theta )\\ -\sin (\theta )& \cos (\theta )\end{array}\right]\left(X-\left[\begin{array}{c}x\\ y\end{array}\right]\right)\right)+\left[\begin{array}{c}x\\ y\end{array}\right]$

Projection onto x-axis

$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

Projection onto y-axis

$\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

Enlargement factor $k>0$ about the origin

$\left[\begin{array}{cc}k& 0\\ 0& k\end{array}\right]$

Shear factor $k>0$ along the x-axis

$\left[\begin{array}{cc}1& k\\ 0& 1\end{array}\right]$

Shear factor $k>0$ along the y-axis

$\left[\begin{array}{cc}1& 0\\ k& 1\end{array}\right]$

Dilation $m$ about y-axis and $n$ about x-axis (note: for a shape to not be dilated, $m$ or $n$ is equal to 1)

$\left[\begin{array}{cc}n& 0\\ 0& m\end{array}\right]$