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Goldman Sachs·Software Engineer·Technical Phone Screen·Junior

Junior
Jun 2026

Summary

Goldman Sachs linear algebra question, pretty much a pure math screen. One system of equations, three unknowns, figure out what kind of solution set you're dealing with. Short and focused.

Questions Asked (1)

Q1

Given the system of three linear equations with three unknowns (2x₁ − 2x₃ = 10, 2x₁ + 3x₂ + x₃ = 20, −x₁ + x₂ + 2x₃ = 5), determine whether the system has a unique solution, infinitely many solutions, no solution, or none of those cases.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

Row reduction is the move here.

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AI HintsAI Generated

Suggested Approach

First, write the system in matrix form Ax = b and compute the determinant of the coefficient matrix. If the determinant is non-zero, the system has a unique solution; if it is zero, analyze the rank of A and the augmented matrix [A|b] to determine whether there are infinitely many solutions or no solution.

Pro tip: After finding the determinant, briefly mention that you could also use Gaussian elimination to check consistency, showing you understand multiple methods. Also, relate this to real-world scenarios like solving linear systems in financial models, which demonstrates practical insight.

1. Write the system in matrix form

Express the equations as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector.

2. Compute the determinant of A

Calculate det(A). If det(A) ≠ 0, the system has a unique solution. If det(A) = 0, proceed to step 3.

3. Check consistency using ranks

If det(A) = 0, compare the rank of A with the rank of the augmented matrix [A|b]. If ranks are equal, there are infinitely many solutions; if not, there is no solution.

4. Conclude and verify

State the conclusion based on the analysis. Optionally, verify by solving the system or checking a specific solution.

Key Points to Mention

  • Matrix representation of linear systems
  • Determinant as a test for unique solution
  • Rank of a matrix and its relation to consistency
  • Augmented matrix and row reduction (Gaussian elimination)
  • Possible outcomes: unique solution, infinitely many solutions, no solution
  • Practical relevance in engineering and finance (e.g., solving linear systems in algorithms)

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.