MA 225 Boston University Linear Algebra equations
Question Description
SHOW ALL OF YOUR WORK
good luck
Tutor Answer
Here's the solution file.:) :)
1. A system of linear equations only has three possibilities of a solution set.
a. What are those three possibilities?
Ans:  There are three types of solutions of the system of linear equations.
• No solution
• Unique solution (one solution)
• Infinitely many solutions
b. Cite the theorem from section 1.6 that states this fact.
“The theorem will be given in section 1.6 of your book”
2. Consider the system of linear equations below of the form Ax =b
𝑥1 + 2𝑥2 + 3𝑥3 = 5
2𝑥1 + 9𝑥2 + 3𝑥3 = 12
𝑥1 + 4𝑥3 = 7
a. Write out the coefficient matrix A.
Sol: 1 2 3
𝐴 = [ 2 9 3]
1 0 4
b. Write the column vector b.
Sol: 5
𝑏 = [12]
7
c. Determine if A is nonsingular by finding the determinant of A. Use either
arrow technique or cofactor expansion along any row or column of A of your
choice.
Sol: Using cofactor expansion along first row to find determinant, we get
1 2 3
𝐴 = 2 9 3
1 0 4
9 3
2 9
2 3
𝐴 = (−1)1+1 (1) 
 + (−1)1+2 (2) 
 + (−1)1+3 (3) 

0 4
1 4
1 0
9 3
2 9
2 3
𝐴 = (−1)2 (1) 
 + (−1)3 (2) 
 + (−1)4 (3) 

0 4
1 4
1 0
9 3
2 9
2 3
𝐴 = 1 
 − 2
 + 3

0 4
1 4
1 0
𝐴 = 1[(9)(4) − (0)(3)] − 2[(2)(4) − (1)(3)] + 3[(2)(0) − (1)(9)]
𝐴 = 1[36 − 0] − 2[8 − 3] + 3[0 − 9]
𝐴 = 1[36] − 2[5] + 3[−9]
𝐴 = 36 − 10 − 27
𝐴 = −1
Hence, matrix A is nonsingular.
d. Is A invertible and why or why not?
Ans:  A matrix is said to be invertible if and only if it has nonzero determinant.
So, according to definition, matrix A is invertible because its determinant is
nonzero.
e. Solve the system of linear equations above by inverting the coefficie...
Brown University
1271 Tutors
California Institute of Technology
2131 Tutors
Carnegie Mellon University
982 Tutors
Columbia University
1256 Tutors
Dartmouth University
2113 Tutors
Emory University
2279 Tutors
Harvard University
599 Tutors
Massachusetts Institute of Technology
2319 Tutors
New York University
1645 Tutors
Notre Dam University
1911 Tutors
Oklahoma University
2122 Tutors
Pennsylvania State University
932 Tutors
Princeton University
1211 Tutors
Stanford University
983 Tutors
University of California
1282 Tutors
Oxford University
123 Tutors
Yale University
2325 Tutors