Final Exam Review:
If you are using your calculator for the matrices arithmetic/calculations, be sure to show your
initial and final matrix.
1. Consider the system of linear equations below:
2𝑥1 + 𝑥2 + 3𝑥3 = 1
4𝑥1 + 3𝑥2 + 5𝑥3 = 1
6𝑥1 + 5𝑥2 + 5𝑥3 = −3
(a) Write the augmented matrix corresponding to this system.
(b) Solve the system using Gauss-Jordan Elimination.
2. Consider the homogenous system of linear equations below:
𝑥1 + 3𝑥2 + 𝑥3 = 0
𝑥3 + 3𝑥4 = 0
(a) Is this system consistent or inconsistent? Explain.
(b) Write the solutions in vector form.
3. Give the solutions of the systems corresponding to the following augmented matrices (already in
reduced row echelon form) below.
1 0 0 2 0 1 −1 3 1 0 3 2
(a) [0 1 0 | −3] (b) [0 0 0 | 0] (c) [0 0 0 | 0]
0 0 0 5 0 0 0 0 0 0 0 0
4. Given the following matrix, find its inverse by hand. An answer without supporting work is worth
zero points. Show your row operations, but you do not have to show the arithmetic.
3 3 −16
𝐴 = [1 1 −6 ]
1 2 −14
5. Find a matrix X such that AX = B for the following. Be sure to show what you are typing
into your calculator. Note: Matrix A is the same matrix from question 4. You do not have to
do any arithmetic by hand or show the elementary row operations.
3 3 −16 1 4 6
𝐴 = [1 1 −6 ] and 𝐵 = [3 1 2]
1 2 −14 2 5 3
6. Given the n + 1 data points below, find the nth degree polynomial that fits those points. Be
sure to show your initial and final matrices.
(2, 25); (1, 0); (0, 5); (-1, 16); (-2, 81)
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, 7. Find the determinant of the following matrices by hand. An answer without supporting work
is worth zero points.
𝑥−𝑦 𝑥 + 2𝑦
(a) 𝐴 = [ ]
𝑥 𝑥−𝑦
2 1 5
(b) 𝐴 = [1 3 2] Use expansion by cofactors.
3 4 6
8. Given that |𝐴| = 15 find the determinants of the following matrices:
(a) B which comes from multiplying R1 in A by 4
(b) B comes from swapping R1 and R5 in A
(c) B which comes from performing 3R1 + R2 = R2 in A
𝑎 𝑏
9. Prove that all nonzero matrices of the form [ ] have an inverse.
−𝑏 𝑎
10. Determine whether the following subset of ℝ3 is a subspace.
𝑊 = {(𝑥, 𝑦, 𝑧) ∈ ℝ4 } where 𝑥 + 𝑦 + 𝑧 = 0
11. Prove if u, v, and w are linearly independent vectors, then the vectors 𝒙𝟏 , 𝒙𝟐 , and 𝒙𝟑 defined
by the following are also linearly independent.
𝒙𝟏 = 𝒖
𝒙𝟐 = 𝒖 + 3𝒗
𝒙𝟑 = 𝒖 + 3𝒗 + 5𝒘
12. Determine whether the following vectors form a basis for ℝ3 . Justify your reasoning.
(a) {(1,0,0), (1,1,0)}
(b) {(1,0,0), (1,1,0), (2,1,0)}
(c) {(1,0,0), (1,1,0), (1,1,1)}
13. Given the following system of equations with coefficient matrix A:
−2𝑥1 + 2𝑥2 + 4𝑥3 = 0
4𝑥1 − 6𝑥2 − 8𝑥3 = 0
(a) Find a basis for the column space of A.
(b) Find a basis for the row space of A.
(c) What is the rank of A?
(d) Find a basis for the solution space of A.
(e) What is the dimension of the null space of A?
This study source was downloaded by 100000901307859 from CourseHero.com on 10-10-2025 06:27:16 GMT -05:00
https://www.coursehero.com/file/251685449/220prfipdf/
If you are using your calculator for the matrices arithmetic/calculations, be sure to show your
initial and final matrix.
1. Consider the system of linear equations below:
2𝑥1 + 𝑥2 + 3𝑥3 = 1
4𝑥1 + 3𝑥2 + 5𝑥3 = 1
6𝑥1 + 5𝑥2 + 5𝑥3 = −3
(a) Write the augmented matrix corresponding to this system.
(b) Solve the system using Gauss-Jordan Elimination.
2. Consider the homogenous system of linear equations below:
𝑥1 + 3𝑥2 + 𝑥3 = 0
𝑥3 + 3𝑥4 = 0
(a) Is this system consistent or inconsistent? Explain.
(b) Write the solutions in vector form.
3. Give the solutions of the systems corresponding to the following augmented matrices (already in
reduced row echelon form) below.
1 0 0 2 0 1 −1 3 1 0 3 2
(a) [0 1 0 | −3] (b) [0 0 0 | 0] (c) [0 0 0 | 0]
0 0 0 5 0 0 0 0 0 0 0 0
4. Given the following matrix, find its inverse by hand. An answer without supporting work is worth
zero points. Show your row operations, but you do not have to show the arithmetic.
3 3 −16
𝐴 = [1 1 −6 ]
1 2 −14
5. Find a matrix X such that AX = B for the following. Be sure to show what you are typing
into your calculator. Note: Matrix A is the same matrix from question 4. You do not have to
do any arithmetic by hand or show the elementary row operations.
3 3 −16 1 4 6
𝐴 = [1 1 −6 ] and 𝐵 = [3 1 2]
1 2 −14 2 5 3
6. Given the n + 1 data points below, find the nth degree polynomial that fits those points. Be
sure to show your initial and final matrices.
(2, 25); (1, 0); (0, 5); (-1, 16); (-2, 81)
This study source was downloaded by 100000901307859 from CourseHero.com on 10-10-2025 06:27:16 GMT -05:00
https://www.coursehero.com/file/251685449/220prfipdf/
, 7. Find the determinant of the following matrices by hand. An answer without supporting work
is worth zero points.
𝑥−𝑦 𝑥 + 2𝑦
(a) 𝐴 = [ ]
𝑥 𝑥−𝑦
2 1 5
(b) 𝐴 = [1 3 2] Use expansion by cofactors.
3 4 6
8. Given that |𝐴| = 15 find the determinants of the following matrices:
(a) B which comes from multiplying R1 in A by 4
(b) B comes from swapping R1 and R5 in A
(c) B which comes from performing 3R1 + R2 = R2 in A
𝑎 𝑏
9. Prove that all nonzero matrices of the form [ ] have an inverse.
−𝑏 𝑎
10. Determine whether the following subset of ℝ3 is a subspace.
𝑊 = {(𝑥, 𝑦, 𝑧) ∈ ℝ4 } where 𝑥 + 𝑦 + 𝑧 = 0
11. Prove if u, v, and w are linearly independent vectors, then the vectors 𝒙𝟏 , 𝒙𝟐 , and 𝒙𝟑 defined
by the following are also linearly independent.
𝒙𝟏 = 𝒖
𝒙𝟐 = 𝒖 + 3𝒗
𝒙𝟑 = 𝒖 + 3𝒗 + 5𝒘
12. Determine whether the following vectors form a basis for ℝ3 . Justify your reasoning.
(a) {(1,0,0), (1,1,0)}
(b) {(1,0,0), (1,1,0), (2,1,0)}
(c) {(1,0,0), (1,1,0), (1,1,1)}
13. Given the following system of equations with coefficient matrix A:
−2𝑥1 + 2𝑥2 + 4𝑥3 = 0
4𝑥1 − 6𝑥2 − 8𝑥3 = 0
(a) Find a basis for the column space of A.
(b) Find a basis for the row space of A.
(c) What is the rank of A?
(d) Find a basis for the solution space of A.
(e) What is the dimension of the null space of A?
This study source was downloaded by 100000901307859 from CourseHero.com on 10-10-2025 06:27:16 GMT -05:00
https://www.coursehero.com/file/251685449/220prfipdf/
