50 pts.
Problem 1. In each part you are given the augmented matrix of a system
of linear equations, with the coefficient matrix in reduced row
echelon form. Determine if the system is consistent and, if it is consistent, find
all solutions.
A.






B.






C.
1
0
0
0
0
0 −1
1
2
0
0
0
0
0
0

0
3 10
0 −5 −1 

3 
1
1

0
0
0 
0
0
0
1
0
0
0
0
0 −1
1
2
0
0
0
0
0
0

0
3
0
0 −5 −4 

1
1
1 

0
0
1 
0
0
0






1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
3
1
5
0
0






Problem 2. Solve the linear system below. Use your calculator to find the
RREF, but write down the augmented matrix and the matrix
you wind up with, and then find all solutions.
2x1 + 2x2 − 2x3 = 8
x2 + x3 = 1
2x1 + x2 − 3x3 = 7
40 pts.
Problem 3. Use row operations to determine if the matrix A is invertible
and, if so, to find the inverse. Show the indivdual row operations,
one by one. You can use a calculator to do the row operations if you wish. Give
the matrix entries in fractional form.
2 4
A=
1 1
1
45 pts.
Problem 4. Consider row operations on matrices with 3 rows.
Recall that for each row operation there is a corresponding elementary matrix
E so that EA is the same as the matrix obtained by applying the row operation
to A.
A. Consider the row operation R2 ↔ R3 .
i.) Find the corresponding elementary matrix E.
ii.) Find the inverse row operation.
iii.) Find the inverse of the matrix E in the first part.
B. Consider the row operation R3 ← 5R3 .
i.) Find the corresponding elementary matrix E.
ii.) Find the inverse row operation.
iii.) Find the inverse of the matrix E in the first part.
C. Consider the row operation R1 ← R1 − 3R3
i.) Find the corresponding elementary matrix E.
ii.) Find the inverse row operation.
iii.) Find the inverse of the matrix E in the first part.
40 pts.
Problem 5. Consider the matrix

1
A = −2
3

−1 2
0 1
1 2
A. Find the cofactors A21 ,A22 and A23 .
B. Show how to compute det(A) by using a cofactor expansion along some row
or column.
40 pts.
Problem 6. Use the method of elimination (i.e., row operations) to find
the determinant of the matrix
5 4
A=
2 6
You can use a calculator to do the row operations, but show the row operations you apply and the intermediate matrices. Sorry, no credit for finding the
determinant by another method!
2
EXAM
Exam 1
Math 2360, Spring 2010
Feb. 16, 2011
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
You may leave when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 6 problems. There are 215 points
total.
Good luck!