BTCS-110 Applied Mathematics Comprehensive
QUESTION Bank | 350+ Most Tested QUESTIONs with
Verified Answers & Step-by-Step Solutions
QUESTION 1
3 5
What is the determinant of the 2 × 2 matrix 𝐴 = [ ]?
2 4
• A. 2
• B. 22
• C. 10
• D. 20
Correct Answer: A. 2
𝑎 𝑏
Detailed Rationale: The determinant of a 2 × 2 matrix [ ] is given
𝑐 𝑑
by 𝑎𝑑 − 𝑏𝑐. Here, (3)(4) − (5)(2) = 12 − 10 = 2.
QUESTION 2
If 𝐴 is a square matrix of order 3 such that |𝐴| = 5, what is the value of
|2𝐴|?
• A. 10
• B. 25
• C. 40
• D. 15
Correct Answer: C. 40
,Detailed Rationale: For an 𝑛 × 𝑛 matrix 𝐴, the scalar multiplication
property of determinants is |𝑘𝐴| = 𝑘 𝑛 |𝐴|. Here, order 𝑛 = 3, scalar
𝑘 = 2, and |𝐴| = 5. Thus, |2𝐴| = 23 ⋅ |𝐴| = 8 ⋅ 5 = 40.
QUESTION 3
1 2
What is the inverse of the matrix 𝐴 = [ ]?
3 4
−2 1
• A. [ ]
3/2 −1/2
−4 2
• B. [ ]
3 −1
1 −2
• C. [ ]
−3 4
2 −1
• D. [ ]
−3/2 1/2
−2 1
Correct Answer: A. [ ]
3/2 −1/2
𝑎 𝑏
Detailed Rationale: The inverse of a 2 × 2 matrix [ ] is
𝑐 𝑑
1 𝑑
−𝑏
[ ]. Here, determinant is (1)(4) − (2)(3) = −2. Thus,
𝑎𝑑−𝑏𝑐 −𝑐
𝑎
1 4 −2 −2 1
𝐴−1 = [ ]=[ ].
−2 −3 1 3/2 −1/2
QUESTION 4
Which of the following matrices is symmetric?
1 2
• A. [ ]
3 1
0 1
• B. [ ]
−1 0
, 1 2
• C. [ ]
2 5
0 0
• D. [ ]
1 0
1 2
Correct Answer: C. [ ]
2 5
Detailed Rationale: A matrix is symmetric if it is equal to its transpose
(𝐴 = 𝐴𝑇 ). For matrix C, the element at row 1, column 2 is 2, and the
element at row 2, column 1 is 2, satisfying 𝑎𝑖𝑗 = 𝑎𝑗𝑖 .
QUESTION 5
2 4
What are the eigenvalues of the upper triangular matrix 𝐴 = [ ]?
0 5
• A. 2 and 5
• B. 4 and 0
• C. 2 and 4
• D. −2 and −5
Correct Answer: A. 2 and 5
Detailed Rationale: The eigenvalues of a triangular (upper or lower) or
diagonal matrix are simply the entries along its main diagonal. Thus, the
eigenvalues are 2 and 5.
QUESTION 6
According to the Cayley-Hamilton Theorem, every square matrix
satisfies:
• A. Its own characteristic equation
• B. The identity matrix equation
QUESTION Bank | 350+ Most Tested QUESTIONs with
Verified Answers & Step-by-Step Solutions
QUESTION 1
3 5
What is the determinant of the 2 × 2 matrix 𝐴 = [ ]?
2 4
• A. 2
• B. 22
• C. 10
• D. 20
Correct Answer: A. 2
𝑎 𝑏
Detailed Rationale: The determinant of a 2 × 2 matrix [ ] is given
𝑐 𝑑
by 𝑎𝑑 − 𝑏𝑐. Here, (3)(4) − (5)(2) = 12 − 10 = 2.
QUESTION 2
If 𝐴 is a square matrix of order 3 such that |𝐴| = 5, what is the value of
|2𝐴|?
• A. 10
• B. 25
• C. 40
• D. 15
Correct Answer: C. 40
,Detailed Rationale: For an 𝑛 × 𝑛 matrix 𝐴, the scalar multiplication
property of determinants is |𝑘𝐴| = 𝑘 𝑛 |𝐴|. Here, order 𝑛 = 3, scalar
𝑘 = 2, and |𝐴| = 5. Thus, |2𝐴| = 23 ⋅ |𝐴| = 8 ⋅ 5 = 40.
QUESTION 3
1 2
What is the inverse of the matrix 𝐴 = [ ]?
3 4
−2 1
• A. [ ]
3/2 −1/2
−4 2
• B. [ ]
3 −1
1 −2
• C. [ ]
−3 4
2 −1
• D. [ ]
−3/2 1/2
−2 1
Correct Answer: A. [ ]
3/2 −1/2
𝑎 𝑏
Detailed Rationale: The inverse of a 2 × 2 matrix [ ] is
𝑐 𝑑
1 𝑑
−𝑏
[ ]. Here, determinant is (1)(4) − (2)(3) = −2. Thus,
𝑎𝑑−𝑏𝑐 −𝑐
𝑎
1 4 −2 −2 1
𝐴−1 = [ ]=[ ].
−2 −3 1 3/2 −1/2
QUESTION 4
Which of the following matrices is symmetric?
1 2
• A. [ ]
3 1
0 1
• B. [ ]
−1 0
, 1 2
• C. [ ]
2 5
0 0
• D. [ ]
1 0
1 2
Correct Answer: C. [ ]
2 5
Detailed Rationale: A matrix is symmetric if it is equal to its transpose
(𝐴 = 𝐴𝑇 ). For matrix C, the element at row 1, column 2 is 2, and the
element at row 2, column 1 is 2, satisfying 𝑎𝑖𝑗 = 𝑎𝑗𝑖 .
QUESTION 5
2 4
What are the eigenvalues of the upper triangular matrix 𝐴 = [ ]?
0 5
• A. 2 and 5
• B. 4 and 0
• C. 2 and 4
• D. −2 and −5
Correct Answer: A. 2 and 5
Detailed Rationale: The eigenvalues of a triangular (upper or lower) or
diagonal matrix are simply the entries along its main diagonal. Thus, the
eigenvalues are 2 and 5.
QUESTION 6
According to the Cayley-Hamilton Theorem, every square matrix
satisfies:
• A. Its own characteristic equation
• B. The identity matrix equation
