Another service from Omega

Problems on Systems of Linear Equations and Matrices


*****

Problem1:

Find the augmented matrix for the following system of linear equations

> ;

                      v + 2 w - y + z = 1

                          3 w + x - z = 2

                              x + 7 y = 1

Solution1

Problem2:

Find the system of linear equations corresponding to the augmented matrix

> ;

                             [3     0    -2     5]
                             [                   ]
                             [7     1     4    -3]
                             [                   ]
                             [0    -2     1     7]

Solution2

Problem3:

The curve,

> ;

                                     2
                              y = a x  + b x + c

passes through the points (x1,y1), (x2,y2), (x3,y3). Show that the coefficients a,b, and c are a solution of the system of linear equations whose augmented matrix is,

> ;

                            [  2                 ]
                            [x1     x1    1    y1]
                            [                    ]
                            [  2                 ]
                            [x2     x2    1    y2]
                            [                    ]
                            [  2                 ]
                            [x3     x3    1    y3]

Solution3

Problem4:

Find the solution to the system of equations with augmented matrix given by,

> ;

                           [1    0    0    -7     8]
                           [                       ]
                           [0    1    0     3     2]
                           [                       ]
                           [0    0    1     1    -5]

Solution4

Problem5:

Use Gauss-Jordan elimination to find the solution to:

> ;

                              5 x - 2 y + 6 z = 0

                               -2 x + y + 3 z = 1

Solution5

Problem6:

Find coefficients a,b,c, and d so that the curve,

> ;

                           2      2
                        a x  + a y  + b x + c y + d = 0

goes through the points: (-4,5), (-2,7) and (4,-3)

Plot this curve. Solution6

Problem7:

Find the inverse of

> ;

                              [cos(t)     sin(t)]
                              [                 ]
                              [-sin(t)    cos(t)]

Solution7

Problem8:

Let A, B, and 0 be 2x2 matrices. Assuming that A is invertible, find a matrix C so that,

> ;

                                 [ -1       ]
                                 [A       0 ]
                                 [          ]
                                 [        -1]
                                 [ C     A  ]

is the inverse of the partitioned matrix

> ;

                                   [A    0]
                                   [      ]
                                   [B    A]

Solution8

Problem9:

Use the result of the previous problem to find the inverse of the matrix,

> ;

                             [ 1    1     0    0]
                             [                  ]
                             [-1    1     0    0]
                             [                  ]
                             [ 1    1     1    1]
                             [                  ]
                             [ 1    1    -1    1]

Solution9

Problem10:

Which of the following are elementary matrices?

> (a);

                                   [ 1    0]
                                   [       ]
                                   [-5    1]
> (b);
                                 [0    0    1]
                                 [           ]
                                 [0    1    0]
                                 [           ]
                                 [1    0    0]
> (c);
                              [2    0    0    2]
                              [                ]
                              [0    1    0    0]
                              [                ]
                              [0    0    1    0]
                              [                ]
                              [0    0    0    1]

Solution10

Problem11:

Consider the matrices,

> ;

                                  [3     4     1]
                                  [             ]
                             A := [2    -7    -1]
                                  [             ]
                                  [8     1     5]
> ;
                                  [3     4     1]
                                  [             ]
                             B := [2    -7    -1]
                                  [             ]
                                  [2    -7    -3]

Find elementary matrices E and F such that,

> ;

                                    E A = B

                                    F B = A

Solution11

Problem12:

Find the inverse of each of the following matrices, where a,b,c,d, and e are all nonzero,

> ;

                                 [a    0    0    0]
                                 [                ]
                                 [0    b    0    0]
                            A := [                ]
                                 [0    0    c    0]
                                 [                ]
                                 [0    0    0    d]


                                 [0    0    0    a]
                                 [                ]
                                 [0    0    b    0]
                            B := [                ]
                                 [0    c    0    0]
                                 [                ]
                                 [d    0    0    0]


                                 [e    0    0    0]
                                 [                ]
                                 [1    e    0    0]
                            C := [                ]
                                 [0    1    e    0]
                                 [                ]
                                 [0    0    1    e]

Solution12

Problem13:

Find the conditions that b's must satisfy for the system to be consistent

> ;

                               x - 2 y - z  = b1

                           -4 x + 5 y + 2 z = b2

                           -4 x + 7 y + 4 z = b3

Solution13

Problem14:

Consider the matrices,

> ;

                                   [2    1     2]
                                   [            ]
                              A := [2    2    -2]
                                   [            ]
                                   [3    1     1]


                                        [x1]
                                        [  ]
                                   x := [x2]
                                        [  ]
                                        [x3]

Show that the equation Ax=x can be rewritten as (A-I)x=0 and use this result to solve Ax=x for x. Also solve Ax=4x.


Solution14

Problem15:

Let A be a symmetric matrix.
  1. Show that A^2 is symmetric.
  2. Show that 2A^2 - 3A + I is symmetric.


Solution15

Problem16:

A square matrix A is called skew-symmetric if,

> ;

                                     T
                                    A  = -A

Show that
  1. If A is an invertible skew-symmetric matrix, then its inverse is also skew-symmetric.
  2. If A and B are skew-symmetric, then so are A^T, A+B, A-B and kA for any scalar k.
  3. Every square matrix A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.


Solution16

Problem17:

Find an upper triangular matrix A, such that,

> ;

                                 3   [1    30]
                                A  = [       ]
                                     [0    -8]
   

Solution17


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Tue Feb 8 16:40:44 EST 2000