# Water's Home

Just another Life Style

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## Matrices and Vectors

The dimension of the matrix is going to be written as the number of row times the number of columns in the matrix. A vector turns out to be a special case of a matrix. Matrix with just one column is what we call a vector.

$\begin{bmatrix} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{bmatrix} + \begin{bmatrix} 4 & 0.5 \\ 2 & 5 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0.5 \\ 4 & 10 \\ 3 & 2 \end{bmatrix}$

#### Scalar Multiplication

$3 * \begin{bmatrix} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 3 & 0 \\ 6 & 15 \\ 9 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{bmatrix} * 3$

## Matrix Vector Multiplication

$\begin{bmatrix} 1 & 3 \\ 4 & 0 \\ 2 & 1 \end{bmatrix} * \begin{bmatrix} 1 \\ 5 \end{bmatrix} = \begin{bmatrix} 16 = 1 * 1 + 3 * 5 \\ 4 = 4 * 1 + 0 * 5 \\ 7 = 2 * 1 + 1 * 5 \end{bmatrix}$

## Matrix Matrix Multiplication

$\begin{bmatrix} C0 & C1 \\ C2 & C3 \end{bmatrix} = \begin{bmatrix} A0 & A1 \\ A2 & A3 \end{bmatrix} * \begin{bmatrix} B0 & B1 \\ B2 & B3 \end{bmatrix}$

$C0 = A0 * B0 + A1 * B2$ $C1 = A0 * B1 + A1 * B3$ $C2 = A2 * B0 + A3 * B2$ $C3 = A2 * B1 + A3 * B3$

## Matrix Multiplication Properties

$A * B \neq B * A$ $A * (B * C) = (A * B) * C$ Identity matrix, has the property that it has ones along the diagonals, right, and so on and is zero everywhere else. $AA^{-1} = A^{-1}A = I$ $AI = IA = A$

## Inverse and Transpose

#### Inverse

$A^{-1}$

#### Transpose

$A^{T}$

$\begin{vmatrix} a & b\\ c & d\\ e & f \end{vmatrix} ^{T} = \begin{vmatrix} a & c & e\\ b & d & f \end{vmatrix}$