Matrices

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Section 5.3

• Matrices
• And
• Systems of Equations

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Systems of Equations in Two Variables
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Matrices

• A rectangular array of numbers is called a matrix
  (plural, matrices).
• Example
• The matrix shown above is an augmented matrix
  because it contains not only the coefficients but
  also the constant terms.
• The matrix is called the coefficient
  matrix.

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Matrices continued

• The rows of a matrix are horizontal.
• The columns of a matrix are vertical.
• The matrix shown has 2 rows and 3 columns.
• A matrix with m rows and n columns is said to be
  of order m ? n.
• When m n the matrix is said to be square.

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Gaussian Elimination with Matrices

• Row-Equivalent Operations
• 1. Interchange any two rows.
• 2. Multiply each entry in a row by the same
• nonzero constant.
• 3. Add a nonzero multiple of one row to
• another row.

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Example

• Solve the following system
• 
  .

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Example continued
First, we write the augmented matrix, writing 0
for the missing y-term in the last equation.
Our goal is to find a row-equivalent matrix of
the form
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Example continued
New row 1 row 2 New row 2 row 1
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Example continued

• We multiply the first row by ?2 and add it to the
  second row.
• We also multiply the first row by ?4 and add it
  to the third row.

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Example continued

• We multiply the second row by 1/5 to get a 1 in
  the second row, second column.

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Example continued

• We multiply the second row
• by ?12 and add it to the
• third row.

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Example continued

• Now, we can write the
• system of equations
• that corresponds to our
• last matrix.

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Example continued

• We back-substitute 3 for z in equation (2) and
  solve for y.

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Example continued

• Next, we back-substitute ?1 for y and 3 for z in
  equation (1) and solve for x.
• The triple (2, ?1, 3) checks in the original
  system of equations, so it is the solution.

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Row-Echelon Form

• 1. If a row does not consist entirely of 0s,
  then the first nonzero element in the row is a 1
  (called a leading 1).
• 2. For any two successive nonzero rows, the
  leading 1 in the lower row is farther to the
  right than the leading 1 in the higher row.
• 3. All the rows consisting entirely of 0s are at
  the bottom of the matrix.
• If a fourth property is also satisfied, a matrix
  is said to be in reduced row-echelon form
• 4. Each column that contains a leading 1 has 0s
  everywhere else.

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Example

• Which of the following matrices are in
  row-echelon form?
• a) b)
• c) d)

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Gauss-Jordan Elimination

• We perform row-equivalent operations on a matrix
  to obtain a row-equivalent matrix in row-echelon
  form. We continue to apply these operations until
  we have a matrix in reduced row-echelon form.
• 
  .

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Gauss-Jordan Elimination Example

• Example Use Gauss-Jordan elimination to solve
  the system of equations from the previous example.

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Gauss-Jordan Elimination continued

• We continue to perform
• row-equivalent operations until we have a matrix
  in reduced row-echelon form.

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Gauss-Jordan Elimination continued

• Next, we multiply the second row by 3 and add it
  to the first row.

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Gauss-Jordan Elimination continued

• Writing the system of equations that corresponds
  to this matrix, we have

We can actually read the solution, (2, ?1, 3),
directly from the last column of the reduced
row-echelon matrix.
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Special Systems

• When a row consists entirely of 0s, the
  equations are dependent and the system is
  equivalent.

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Special Systems

• When we obtain a row whose only nonzero entry
  occurs in the last column, we have an
  inconsistent system of equations.
• For example, in the matrix
• the last row corresponds to the false
  equation
• 0 9, so we know the original system has no
  solution.

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Another Example

• Solve the system of equations using Gaussian
  elimination.
• x 3y 6z 7
• 2x y 2z 0
• x y 2z -1

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Yet Another Example

• Solve the system of equations using Gaussian
  elimination.
• x 2y 1
• 2x 4y 3