Definition: A matrix is a rectangular array of numbers (or other mathematical objects).
Example: $\left[\begin{array}{cc} 2 & -1 \\ 7 & 3 \end{array}\right]$ is a $2 \times 2$ matrix.
Example: $\left[\begin{array}{ccc} -1 & -5 & 1\\ 0 & 3 & 6 \end{array}\right]$ is a $2 \times 3$ matrix.
Example: $\left[\begin{array}{cc} 4 & 3 \\ 1 & 3 \\ -8 & 5 \\\end{array}\right]$ is a $3 \times 2$ matrix.
Matrices as Representations of Systems
Recall: a $2 \times 2$ system of equations means 2 equations in 2 unknowns.
Example: $$\begin{array}{l}-2x+3y=4\\2x-2y=6\\\end{array}$$ A Shorthand Notation: we may rewrite the above in matrix form: $$\left[\begin{array}{cc|c} -2 & 3 & 4 \\ 2 & -2 & 6\\\end{array}\right]$$
Matrices as Representations of Systems
Examples: Rewrite the following systems of equations in augmented matrix form. $$\left\{\begin{array}{c}5x-9y=0\\7x+y=8\\\end{array}\right\}$$ $$\left\{\begin{array}{c} -3y=9\\6x+4y=5\\\end{array}\right\}$$ $$\left\{\begin{array}{c}x=9\\y=8\\\end{array}\right\}$$
Matrices as Representations of Systems
Examples: Rewrite the augmented matrices as a system of equations. $$\left[\begin{array}{cc|c} 1 & 3 & 7 \\ 5 & -4 & 8 \\\end{array}\right]$$ $$\left[\begin{array}{cc|c} 2 & -5 & 3 \\ 6 & 0 &7\\\end{array}\right]$$ $$\left[\begin{array}{cc|c} 1 & 0 & 5 \\0 & 1 &9\\\end{array}\right]$$
Question: Why do we go to all this trouble to invent a new notation for systems?
Answer: Because doing algebra with matrices is simpler and cleaner.
Solving Systems
Example: For old time's sake, let's solve the system of equations using the addition (elimination) method. $$\begin{array}{l}-2x+3y=4\\2x-2y=6\\\end{array}$$
Elementary Row Operations
Algebraic Operation
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Elementary Row Operation
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Solving Systems
Example: Now solve the system of equations performing "elementary row operations" on its augmented matrix. $$\begin{array}{l}-2x+3y=4\\2x-2y=6\\\end{array}$$
So, As You Can See...
If you can massage the augmented matrix of a system into the form $$\left[\begin{array}{cc|c} 1 & 0 & \mbox{number}_1 \\0 & 1 & \mbox{number}_2\\\end{array}\right]$$ by elementary row operations, you have effectively solved the system. I.e., $$\begin{array}{c} x=\mbox{number}_1 \\ y=\mbox{number}_2 \\ \end{array}$$
More Examples: Solve the systems of equations by performing elementary row operations on their corresponding augmented matrices. $$\left\{\begin{array}{l}6x-4y=2\\-3x+3y=-9\\\end{array} \right\} \,\,\,\, \mbox{one solution}$$ $$\left\{\begin{array}{l}-18x-6y=-18\\-3x-y=-3\\\end{array} \right\} \,\,\,\, \mbox{infinite solutions}$$ $$\left\{\begin{array}{l}4x-2y=6\\2x-y=2\\\end{array} \right\} \,\,\,\, \mbox{no solutions}$$
Application: A goldsmith named would like to make 85 g of a gold alloy which is 75% gold. How much of an alloy which is 82% gold, and another alloy which is 55% gold, should the goldsmith use? Round your answer to the nearest gram.