UTMA Linear Systems and Matrices 1 — Questions and Answers
Question 1: What is the solution to the system: x + y = 5 and x − y = 1?
- x=2, y=3
- x=3, y=2 (Correct answer)
- x=4, y=1
- x=1, y=4
Correct answer: x=3, y=2
Adding the equations: 2x = 6 → x = 3; substituting: y = 5 − 3 = 2.
Question 2: What is the determinant of the matrix [[2, 3],[1, 4]]?
- 5 (Correct answer)
- 8
- 11
- −5
Correct answer: 5
det = 2×4 − 3×1 = 8 − 3 = 5.
Question 3: Which method eliminates a variable by multiplying equations?
- Substitution
- Graphing
- Elimination (Correct answer)
- Matrix inversion
Correct answer: Elimination
The elimination method multiplies equations by constants to cancel a variable when added.
Question 4: How many solutions does the system y = 2x + 1 and y = 2x − 3 have?
- None (Correct answer)
- Exactly one
- Exactly two
- Infinitely many
Correct answer: None
Both lines have slope 2 but different y-intercepts, so they are parallel and never intersect.
Question 5: What is the product of matrices A = [[1,0],[0,1]] and B = [[3,4],[5,6]]?
- [[3,4],[5,6]] (Correct answer)
- [[1,0],[0,1]]
- [[4,4],[5,7]]
- [[8,10],[5,6]]
Correct answer: [[3,4],[5,6]]
The identity matrix I times any matrix B equals B.
Question 6: In Gaussian elimination, what is the goal of row operations?
- To swap all rows
- To convert the matrix to row echelon form (Correct answer)
- To multiply all entries by −1
- To add rows randomly
Correct answer: To convert the matrix to row echelon form
Gaussian elimination uses row operations to reduce the augmented matrix to row echelon form for easy back-substitution.
UTMA Linear Systems and Matrices 1
What is the solution to the system: x + y = 5 and x − y = 1?