Finish writing up notes on matrices lecture 1
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@ -124,7 +124,7 @@ always hold true as there are many solutions.
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- The exponential function $f(x) = \exp x$ may be wirtten as an infinite series:
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$$\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... $$
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$$\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots $$
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- The function $f(x) = e^{-x}$ is just $\frac 1 {e^x}$
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- Note the important properties:
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@ -391,7 +391,7 @@ $$z^3 = 8i$$
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</details>
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# Matrices (and Simultaneous Equations)
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# Systems of Equations (Simultaneous Equations)
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## Gaussian Elimination
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@ -421,7 +421,7 @@ a = 0, b \ne 0 &\rightarrow \text{no solution for $x$} \\
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a = 0, b = 0 &\rightarrow \text{infinite solutions for $x$}
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\end{align*}
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### 2x2 System
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### 2x2 Systems
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A 2x2 system is one with 2 equations and 2 unknown variables.
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@ -457,7 +457,7 @@ You can check the values for $x_1$ and $x_2$ are correct by substituting them in
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</details>
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### 3x3 System
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### 3x3 Systems
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A 3x3 system is one with 3 equations and 3 unknown variables.
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@ -496,3 +496,290 @@ These values can be back-substituted into any of the first 3 equations to find o
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\end{align*}
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</details>
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<details>
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<summary>
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#### Example 2
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\begin{align*}
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x_1 + x_2 - 2x_3 &= 1 &R_1 \\
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2x_1 - x_2 - x_3 &= 1 &R_2 \\
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x_1 + 4x_2 + 7x_3 &= 2 &R_3 \\
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\end{align*}
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</summary>
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1. Eliminate $x_1$ from $R_2$, $R_3$:
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\begin{align*}
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x_1 + x_2 - 2x_3 &= 1 &R_1' = R_1\\
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- 3x_2 - 5x_3 &= -1 &R_2' = R_2 - 2R_1 \\
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3x_2 + 5x_3 &= 1 &R_3' = R_3 - R_1 \\
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\end{align*}
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We've created another 2x2 system of $R_2'$ and $R_3'$
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2. Eliminate $x_2$ from $R_3''$
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\begin{align*}
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x_1 + x_2 - 2x_3 &= 1 &R_1'' = R_1' = R_1\\
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- 3x_2 - 5x_3 &= -1 &R_2'' = R_2' = R_2 - 2R_1 \\
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0x_3 &= 0 &R_3'' = R_3 '+ R_2' \\
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\end{align*}
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We can see that $x_3$ can be any number, so there are infinite solutions. Let:
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$$x_3 = t$$
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where $t$ can be any number
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3. Substitute $x_3$ into $R_2''$:
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$$R_2'' = -3x_2 - 5t = -1 \rightarrow x_2 = \frac 1 3 - \frac{5t} 3$$
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4. Substitute $x_2$ and $x_3$ into $R_1''$:
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$$R_1'' = x_1 + \frac 1 3 - \frac{5t} 3 + 2t = 1 \rightarrow x_1 = \frac 2 3 - \frac t 3$$
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</details>
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## Systems of Equations and Matrices
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Many problems in engineering have a very large number of unknowns and equations to solve
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simultaneously.
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We can use matrices to solve these efficiently.
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Take the following simultaneous equations::
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\begin{align*}
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3x_1 + 4x_2 &= 2 &\text{(1)} \\
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x_1 + 2x_2 &= 0 &\text{(2)}
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\end{align*}
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They can be represented by the following matrices:
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\begin{align*}
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A &= \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix} \\
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\pmb x &= \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \\
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\pmb b &= \begin{pmatrix} 2 \\ 0 \end{pmatrix} \\
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\end{align*}
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You can then express the system as:
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$$A\pmb x = \pmb b$$
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<details>
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<summary>
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#### A 3x3 System as a Matrix
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</summary>
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\begin{align*}
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2x_1 + 3x_2 - x_3 &= 5 \\
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4x_1 + 4x_2 - 3x_3 &= 3 \\
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2x_1 - 3x_2 + x_3 &= -1
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\end{align*}
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Could be expressed in the form $A\pmb x = \pmb b$ where:
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\begin{align*}
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A &= \begin{pmatrix} 2 & 3 & -1 \\ 4 & 4 & -3 \\ 2 & -3 & -1 \end{pmatrix} \\
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\pmb x &= \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \\
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\pmb b &= \begin{pmatrix} 5 \\ 3 \\ -1 \end{pmatrix} \\
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\end{align*}
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</details>
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<details>
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<summary>
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#### An $m\times n$ System as a Matrix
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</summary>
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\begin{align*}
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a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n &= b_1 \\
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a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n &= b_2 \\
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\cdots \\
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a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n &= b_m \\
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\end{align*}
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Could be expressed in the form $A\pmb x = \pmb b$ where:
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\begin{align*}
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A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\
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a_{21} & a_{22} & \cdots & a_{2n} \\
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\vdots & & & \vdots \\
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a_{m1} & a_{m2} & \cdots & a_{mn}
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\end{pmatrix},
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\pmb x = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix},
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\pmb b = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}
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\end{align*}
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</details>
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# Matrices
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## Order of a Matrix
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The order of a matrix is its size e.g. $3\times2$ or $m\times n$
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## Column Vectors
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- Column vectors are matrices with only one column:
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$$ \begin{pmatrix} 1 \\ 2 \end{pmatrix} \begin{pmatrix} 4 \\ 45 \\ 12 \end{pmatrix} $$
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- Column vector variables typed up or printed are expressed in $\pmb{bold}$ and when it is
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handwritten it is \underline{underlined}:
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$$ \pmb x = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$$
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## Matrix Algebra
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### Equality
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Two matrices are the same if:
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- Their order is the same
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- Their corresponding elements are the same
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### Addition and Subtraction
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Only possible if their order is the same.
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\begin{align*}
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A + B&= \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
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a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
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\vdots & & & \vdots \\
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a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn}
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\end{pmatrix} \\
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A - B&= \begin{pmatrix} a_{11} - b_{11} & a_{12} - b_{12} & \cdots & a_{1n} - b_{1n} \\
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a_{21} - b_{21} & a_{22} - b_{22} & \cdots & a_{2n} - b_{2n} \\
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\vdots & & & \vdots \\
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a_{m1} - b_{m1} & a_{m2} - b_{m2} & \cdots & a_{mn} - b_{mn}
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\end{pmatrix},
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\end{align*}
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### Zero Matrix
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This is a matrix whose elements are all zeros.
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For any matrix $A$,
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$$A + 0 =A$$
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We can only add matrices of the same order, therefore 0 must be of the same order as $A$.
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### Multiplication
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Let
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$$
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\begin{matrix}
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A & m\times n \\
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B & p\times q
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\end{matrix}
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$$
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To be able to multiply $A$ by $B$, $n = p$.
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If $n \ne p$, then $AB$ does not exist.
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$$
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\begin{matrix}
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A & B & = & C \\
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m\times n & p \times q & & m\times q
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\end{matrix}
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$$
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When $C = AB$ exists,
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$$C_{ij} = \sum_r\! a_{ir}b_{rj}$$
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That is, $C_{ij}$ is the 'product' of the $i$th row of $A$ and $j$th column of $B$.
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#### Multiplication of a Matrix by a Scalar
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If $\lambda$ is a scalar, we define
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$$
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\lambda a = \begin{pmatrix} \lambda a_{11} & \lambda a_{12} & \cdots & \lambda a_{1n} \\
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\lambda a_{21} & \lambda a_{22} & \cdots & \lambda a_{2n} \\
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\vdots & & & \vdots \\
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\lambda a_{m1} & \lambda a_{m2} & \cdots & \lambda a_{mn}
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\end{pmatrix},
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$$
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<details>
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<summary>
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#### Example 1
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</summary>
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$$
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\begin{pmatrix} 1 & -1 \\ 2 & 1 \end{pmatrix}
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\begin{pmatrix} 0 & 1 \\ 3 & 2 \end{pmatrix} =
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\begin{pmatrix} -3 & -1 \\ 3 & 4 \end{pmatrix}
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$$
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$$
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\begin{pmatrix} 0 & 1 \\ 3 & 2 \end{pmatrix}
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\begin{pmatrix} 1 & -1 \\ 2 & 1 \end{pmatrix} =
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\begin{pmatrix} 2 & 1 \\ 7 & -1 \end{pmatrix}
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$$
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</details>
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<details>
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<summary>
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#### Example 2
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</summary>
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$$
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A = \begin{pmatrix} 4 & 1 & 6 \\ 3 & 2 & 1 \end{pmatrix},\,
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B = \begin{pmatrix} 1 & 1 \\ 1 & 2 \\ 1 & 0 \end{pmatrix}
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$$
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$$
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AB = \begin{pmatrix} 11 & 6 \\ 6 & 7 \end{pmatrix},\,
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BA = \begin{pmatrix} 7 & 3 & 7 \\ 10 & 5 & 8 \\ 4 & 1 & 6 \end{pmatrix}
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$$
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</details>
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### Other Properties of Matrix Algebra
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- $(\lambda A)B = \lambda(AB) = A(\lambda B)$
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- $A(BC) = (AB)C = ABC$
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- $(A+B)C = AC + BC$
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- $C(A+B) = CA + CB$
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- In general, $AB \ne BA$ even if both exist
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- $AB = 0$ does not always mean $A = 0$ or $B = 0$:
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$$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix}3 & 0 \\ 0 & 0 \end{pmatrix} =
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\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = 0$$
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<details>
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<summary>
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It follows that $AB = AC$ does not imply that $B=C$ as
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$$AB = AC \leftrightarrow A(B + C) = 0$$
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and as $A$ and $(B-C)$ are not necessarily 0, $B$ is not necessarily equal to $C$:
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</summary>
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$$AB = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix} =
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\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$$
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and
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$$AC = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix}1 & 2 \\ 1 & 0 \end{pmatrix} =
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\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = AB$$
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but $B \ne C$
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</details>
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