 # 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 If A is an invertible skew-symmetric matrix, then its inverse is also skew-symmetric. If A and B are skew-symmetric, then so are A^T, A+B, A-B and kA for any scalar k. 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