5'3 Matrices and Systems of Equations

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5.3Matrices and Systems of Equations

• Solve systems of equations using matrices.

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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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Representing a system of equations in a matrix

• If a linear system of 3 equations involved 3
  variables, each column represents the different
  variables constant, and each row represents a
  separate equation.
• Example Write the following system as a matrix
• 2x 3y 3z 7
• 5x y 4z 2
• 4x 2y - z 6

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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)
• Matrices (a) and (d) satisfy the row-echelon
  criteria. In (b) the first nonzero element is not
  1. In (c), the row consisting entirely of 0s is
  not at the bottom of the matrix.

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Write the system for the given matrix. Once the
matrix is written use back-substitution to solve
the system.
Since z 8, replace z in equation 2 to get y-
3(8) 4. Solve to get y 28. Now rewrite
equation 1 using the z and y values and
solve for x. x -6(28) 7(8) 5 gives x
117. The system solution is (117, 28, 8)
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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
• 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
Interchange row 1 and row 2. Next cause entries
under the 1 in column 1 to change to 0s.
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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• We multiply the second row by 1/5 to get a 1 in
  the second row, second column.
• We multiply the second row by ?12 and add it to
  the third row.
• Now, we can write the system
• of equations that corresponds
• to the last matrix above

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Example continued

• We back-substitute 3 for z in equation (2) and
  solve for y.
• 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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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.
• Example Use Gauss-Jordan elimination to solve
  the system of equations from the previous
  example we had obtained the matrix
• 
  .

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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.
• 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.
• 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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Dependent system Infinitely many solutions
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The system is inconsistent. No solution.