PRE CALCULUS Matrices and Vectors 2 — Questions and Answers
Question 1: What is the inverse of the matrix [[2, 1], [5, 3]]?
- [[3, −1], [−5, 2]] (Correct answer)
- [[3, 1], [5, 2]]
- [[2, −1], [−5, 3]]
- [[1, 0], [0, 1]]
Correct answer: [[3, −1], [−5, 2]]
For [[a,b],[c,d]], the inverse is (1/det)[[d,−b],[−c,a]]; det = 1, so inverse = [[3,−1],[−5,2]].
Question 2: Which of the following represents the cross product result for 3D vectors?
- A vector perpendicular to both (Correct answer)
- A scalar
- A zero vector always
- The sum of the vectors
Correct answer: A vector perpendicular to both
The cross product of two 3D vectors produces a vector perpendicular to both original vectors.
Question 3: How many solutions does a system have if its coefficient matrix has determinant = 0?
- Zero or infinitely many (Correct answer)
- Exactly one
- Always infinitely many
- Always zero solutions
Correct answer: Zero or infinitely many
A zero determinant means the matrix is singular, so the system is either inconsistent (no solution) or dependent (infinitely many).
Question 4: What is the scalar projection of u = ⟨4, 3⟩ onto v = ⟨1, 0⟩?
- 4 (Correct answer)
- 3
- 5
- 7
Correct answer: 4
The scalar projection of u onto v is (u · v)/|v| = (4·1 + 3·0)/1 = 4.
Question 5: A matrix A is called symmetric if:
- A = Aᵀ (Correct answer)
- A = A⁻¹
- det(A) = 1
- A has all positive entries
Correct answer: A = Aᵀ
A symmetric matrix equals its own transpose, meaning element aᵢⱼ = aⱼᵢ for all i, j.
Question 6: Which vector has magnitude 1 and same direction as ⟨0, −5⟩?
- ⟨0, −1⟩ (Correct answer)
- ⟨0, 5⟩
- ⟨−1, 0⟩
- ⟨0, −5⟩
Correct answer: ⟨0, −1⟩
The unit vector is obtained by dividing each component by the magnitude: ⟨0, −5⟩/5 = ⟨0, −1⟩.
PRE CALCULUS Matrices and Vectors 2
What is the inverse of the matrix [[2, 1], [5, 3]]?