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
|
> ;
[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.
|
Solution15
Problem16:A square matrix A is called skew-symmetric if, |
> ;
T
A = -A
Show that
|
Solution16
Problem17:Find an upper triangular matrix A, such that, |
> ;
3 [1 30]
A = [ ]
[0 -8]
| Solution17 |