45 lines
1.2 KiB
Markdown
45 lines
1.2 KiB
Markdown
#Math #Algebra
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A matrix is an $n$ by $m$ set of values. A $4 \times 3$can be notated by:
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$$
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\begin{bmatrix}a_1 & a_2 & a_3 \cr b_1 & b_2 & b_3 \cr c_1 & c_2 & c_3 \cr d_1 & d_2 & d_3 \end{bmatrix}
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$$
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To get a value from matrix $a$ in row $r$ and column $c$, use:
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$$
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a_{r, c}
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$$
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# Addition
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With two matrices of the same order, add corresponding elements:
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$$
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\begin{bmatrix} a_1 & b_1 \cr c_1 & d_1 \end{bmatrix} + \begin{bmatrix} a_2 & b_2 \cr c_2 & d_2 \end{bmatrix} = \begin{bmatrix} a_1 + a_2 & b_1 + b_2 \cr c_1 + c_2 & d_1 + d_2 \end{bmatrix}
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$$
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# Subtraction
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With two matrices of the same order, subtract corresponding elements:
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$$
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\begin{bmatrix} a_1 & b_1 \cr c_1 & d_1 \end{bmatrix} - \begin{bmatrix} a_2 & b_2 \cr c_2 & d_2 \end{bmatrix} = \begin{bmatrix} a_1 - a_2 & b_1 - b_2 \cr c_1 - c_2 & d_1 - d_2 \end{bmatrix}
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$$
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# Scalar Multiplication
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When multiplying a matrix by a scalar, multiply each element by said scalar:
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$$
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s\begin{bmatrix} a & b \cr c & d \end{bmatrix} = \begin{bmatrix} sa & sb \cr sc & sd \end{bmatrix}
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$$
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# Matrix Multiplication
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Let $a$ be an $i$ by $j$ matrix and $b$ be a $m$ by $n$ matrix. If $j = m$, $ab$ is defined.
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$$
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ab_{c, d} = \sum _{k = 1}^{j} a_{c, k}b_{k, d}
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$$ |