Matrices-row echelon forms and systems of linear equations

Question

Questions:

Matrices : Row Echelon Forms and Systems of Linear Equations

Row reduce matrix to reduced echelon from. Circle pivot positions in the final matrix and in the original matrix, and list the pivot columns.

1) ( 1  3  5  7 )                      
    ( 3  5  7  9 )                     
    ( 5  7  9  1 )                 

Find the general solutions of the systems whose augmented matrices are given.

2) ( 01 - 65   )                      
    ( 1 -27 - 6 )                     

3) ( 1 - 2 - 13 )                      
    ( 3 - 6 - 22 )

Use the echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine the solution.

4) a) ( . * * * * )                      
         ( 0  .   * * )                     
         ( 0 0 0 .  0)

b) ( 0 . *** )                      
    ( 0 0 . ** )                     
    ( 0 0 0 0 . )

Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.

5) ( 23 h )                      
    (  467 )

6) ( 1 - 3 - 2)
     ( 5h7     )

Choose h and k such that the system has (a) no solution, (b) a unique solution and (c) many solutions. Give separate answers for each part.

7)  x₁ + hx₂ = 2

     4x₁ + 8x₂ = k

8) Mark true or false. Justify answers

a) in some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations

b) the row reduction algorithm applies only to augmented matrices for a linear system

c) a basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix

d) finding a parametric description of the solution set of a linear system is the same as solving the system

e) if one row in an echelon form of an augmented matrix is [0 0 0 5 0], the associated linear system is inconsistent

9) Mark true or false. Justify answers

a) the echelon form of a matrix is unique

b) the pivot positions in a matrix depend on whether row interchanges are used in the row reduction process

c) reducing a matrix to echelon form is called the forward phase of the row reduction process

d) whenever a system has free variables, the solution set contains many solutions

e) a general solution of a system is an explicit description of all solutions of the system

keywords: row-echelon