# MA 225 Boston University Linear Algebra equations

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### Question Description

good luck

Bilal_Mursaleen
School: Purdue University

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 co-factor expansion along any row or column of A of your
choice.
Sol: Using co-factor 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 non-singular.
d. Is A invertible and why or why not?
Ans: - A matrix is said to be invertible if and only if it has non-zero determinant.
So, according to definition, matrix A is invertible because its determinant is
non-zero.
e. Solve the system of linear equations above by inverting the coefficie...

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