(The University Of Choice)
MASINDE MULIRO UNIVERSITY OF
SCIENCE AND TECHNOLOGY
(MMUST)
(MAIN EXAMINATION)
UNIVERSITY EXAMINATIONS
2025/2026 ACADEMIC YEAR
SECOND YEAR FIRST SEMESTER EXAMINATIONS
FOR THE DEGREE
OF
BACHELOR OF SCIENCE AND BACHELOR OF ARTS
COURSE CODE:
MAT 201
COURSE TITLE:
LINEAR ALGEBRA I
DATE:
TIME:
Instructions to candidates:
Answer Question ONE (Compulsory) and ANY other TWO Questions
Time: 2 hours
This Paper Consists of 4 Printed Pages. Please Turn Over.
1
QUESTION ONE (30MARKS) COMPULSORY
a) Given the system of linear equations, use the Gauss-Jordan elimination techneque to
obtain the values of 𝑥1 , 𝑥2 and 𝑥3 .
(4marks)
4𝑥1 + 8𝑥2 − 12𝑥3 = 44
3𝑥1 + 6𝑥2 − 8𝑥3 = 32
−2𝑥1 − 𝑥2 = −7
b) I) state TWO examples of vector spaces
(2marks)
3
ii) Determine if 𝑊 = {(𝑥, 𝑦, 2)|𝑥, 𝑦 ∈ ℝ} is a subspace of ℝ .
(3marks)
1 2 3
c) By first partitioning the matrix 𝐴 = (4 3 2) , find the determinant of A.
(4marks)
1 0 3
d) i)Show that if 𝑆 is a subset of 𝑉, then the set spanned or generated by 𝑆 is a subspace of 𝑉
(3marks)
ii) Determine if (2, −1) is in the set generated by 𝑆 = {(3,1), (2,2)}
(3marks)
e) Consider 𝑇: ℝ2 ⟶ ℝ2 defined as 𝑇(𝑥1 , 𝑥2 ) = (𝑥1 + 1, 𝑥2 ). Determine wether or not 𝑇 is
a linear transformation.
(3marks)
𝜋
𝜋
f) Given that 𝑓(𝑥) = sin 𝑥, sin 2𝑥 = 𝑔(𝑥) and 𝑥0 = 4 . Evaluate W(sin 𝑥 , sin 2𝑥 , 4 )
(4marks)
g) i)Given that 𝐴 and 𝐵 are matrices, simplify (𝐴 + 𝐵)2 and (𝐴 + 𝐵)(𝐴 − 𝐵).
(2marks)
2 1 3
4 −1
ii) Consider 𝐴 = (2 0 1) and 𝐵 = (3 2 ). Evaluate 𝐵 𝑇 𝐴.
(2marks)
1 1 1
2 1
QUESTION TWO (15 MARKS)
a) Write the following matrix into its reduced echelon form.
0 0
𝐴 = [3 3
4 4
2 −2
−3 9
−2 11
(5marks)
2
12 ]
12
b) Determine if the set 𝑆 = {2 + 𝑥 + 𝑥 2 , 𝑥 − 2𝑥 2 , 2 + 3𝑥 − 𝑥 2 } is linearly independent in
the set of polynomials 𝑃2 .
(4marks)
𝑥1
1 2 0 𝑥
c) Let 𝑇: ℝ3 ⟶ ℝ2 be defined by 𝑇(𝑥) = (
) ( 2 ). Find
−1 3 2 𝑥
3
i)
Nullity of 𝑇.
(4marks)
ii)
The rank of 𝑇
(2marks)
QUESTION THREE (15 MARKS)
1
a) Using the cofactors of 𝐴 = (4
1
2 3
3 2). Find the inverse of 𝐴.
0 3
2
(7marks)
b) Find the basis and dimension of the solution space for the system of equations. (8marks)
2𝑥1 + 2𝑥2 − 𝑥3 + 𝑥5 = 0
−𝑥1 − 𝑥2 + 2𝑥3 − 3𝑥4 + 𝑥5 = 0
𝑥1 + 𝑥2 − 2𝑥3 − 𝑥5 = 0
𝑥3 + 𝑥4 + 𝑥5 = 0
QUESTION FOUR (15 MARKS)
a) Solve by determinants the following set of simultaneous equations.
(6marks)
5𝑥 − 6𝑦 + 4𝑧 = 15
7𝑥 + 4𝑦 − 3𝑧 = 19
2𝑥 + 𝑦 − 6𝑧 = 46
2
3
b) Determine if 𝑇: ℝ ⟶ ℝ defined as 𝑇(𝑥1 , 𝑥2 ) = (𝑥1 + 𝑥2 , 2𝑥1 , −𝑥2 ) is a linear
transformation.
(5marks)
c) Demonstrate that there is no solution to the system.
(4marks)
𝑥1 + 𝑥2 + 5𝑥3 = 3
𝑥2 + 3𝑥3 = −1
𝑥1 + 2𝑥2 + 8𝑥2 = 3
QUESTION FIVE (15 MARKS)
𝑥1
2𝑥1 + 𝑥2 − 𝑥3
𝑥
a) Given that 𝑇: ℝ ⟶ ℝ is defined as 𝑇 ( 2 ) = ( −𝑥3 + 𝑥2 ). Find matrix of
𝑥3
−𝑥1 − 𝑥2 − 𝑥3
𝑇.
(3marks)
b) For the vectors 𝑣1 = (1,1, −1), 𝑣2 = (4,0,1) and 𝑣3 = (3, −1,2)
i). Find the basis for the subspace spanned by these vectors and show that the vectors in
the basis are linearly independent.
(6marks)
ii) Write the remaining vectors as a linear combination of the vectors in the basis. .
(1mark)
c) Find the solutions to the homogenous system.
(5marks)
𝑥1 + 𝑥2 − 𝑥4 = 0
𝑥1 + 2𝑥2 − 𝑥3 + 2𝑥4 = 0
−𝑥1 − 2𝑥2 + 2𝑥3 − 3𝑥4 = 0
3
3
3