9.3: Introduction to Matrices

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9.3 Introduction to Matrices Row Reduced
Echelon Form

• The glory of matrices revealed

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Where were headed...

• Two equations, two unknowns doesnt really
  provide much of a challenge anymore. Wed like
  to be able to extend our skillset to larger
  systems of equations. Lets look at a larger
  system

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How to handle

• We still have the same three options graphing,
  substitution, or elimination. But usually,
  elimination is the best choice.
• The key here is to take any two equations and
  simplify in such a way that it goes from 3
  variables down to 2.
• Im going to work with equations 1 and 2 first.

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Solving 3 equations, 3 unknowns.

• Now lets look at equations 1 and 3.

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Whew! Is there another way?

• Yes! Notice that the variables dont really
  matter. Its the coefficients were manipulating
  in order to make stuff disappear. So instead of
  working with the whole equations, lets just work
  with the coefficients in something called a
  matrix.

Coefficient Matrix
Augmented Matrix
Well work with augmented matrices today and look
at coefficient matrices later!
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Watch out!

• Notice that the first column represented the
  coefficients of x, the second column was the
  coefficients of y, the third column was the
  coefficients of z and the fourth column were the
  non-variable coefficients.
• Its important that your coefficients are lined
  up correctly or else youll get a wrong answer!

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Matrix Notation

• Matrices are denoted using double-subscript
  notation.
• aij means matrix a where i row position and j
  column position.
• For example, a23 means 2nd row, 3rd column.

2 x 2 example
3 x 3 example
m x n example
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Square Matrices

• Square matrices m n
• Main Diagonal From top left to bottom right
• The main diagonal is important to us - youll see
  why later.

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Elementary Row Operations

• VERY similar to the theorem on systems of
  equations we talked about yesterday.
• You have three things youre allowed to do inside
  matrices
• Swap any two rows with one another
• Multiply a row by a nonzero constant
• Add rows to together to form a new row that
  replaces an old row

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Symbology

• We can write these rules in mathematical
  shorthand
• The arrow means replaces. The double arrow
  means swaps.
• Ri Rj (Swapping Row i with Row j)
• kRi Ri (Replacing row R i after
  multiplying by a constant)
• kRi Rj Rj (Adding rows together to
  replace a row)

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What we like.

• If we can get a matrix in row reduced echelon
  form, then we know what our variables are equal
  to. So whats row reduced echelon form?
• Ones along the main diagonal. Zeroes everywhere
  else (except for the numbers after the bar.)

This tells us that x a, y b and z c.
Notice that this idea extends to any number of
equations
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Lets rework past example.

• The equations were

So the augmented matrix was
Remember, the goal is to convert this matrix into
a row reduced matrix
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Solution
-4R1 R2 R2
3R1 R3 R3
Notice I now have zeroes under the top left
entry. Thats what I want!
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Lets fix column 2!
1/10R2 R2
2R2 R3 R3
1/4R3 R3
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Finally, fix column 3.

• Notice that I now know z 2.

2R3 R2 R2
2R2 R1 R1
-3R3 R1 R1
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So what were the steps?

• Make a11 1, either by swapping rows or
  multiplying by a constant.
• Use the top row to make the everything under a11
  equal to zero.
• Make a22 1.
• Use row 2 to make the other entries above and
  below equal to zero.
• Make a33 1.
• Use row 3 to make the other entries above equal
  to zero.

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Your turn!

• Solve the system using matrices

Solution