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 GaussJordan 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 (AI)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 skewsymmetric if, 
> ;
T A = A
Show that

Solution16
Problem17:Find an upper triangular matrix A, such that, 
> ;
3 [1 30] A = [ ] [0 8]
Solution17 