Linear Algebra quiz

More questions about Linear Algebra

These visible examples show the question style. LearnClash turns the full topic into ranked duels and spaced repetition.

1. 1 Easy

   What does a square matrix with determinant 0 lack?

   Answer: Inverse

   Why: Because $\det(AB) = \det A \cdot \det B$ and $\det I = 1$, an inverse would need 0 times its determinant to equal 1.

2. 2 Medium

   Which number always equals the sum of a matrix's eigenvalues?

   Answer: Trace

   Why: The trace is just the sum of the diagonal entries, while multiplying the eigenvalues gives the determinant.

3. 3 Hard

   Where did Gaussian elimination appear long before Gauss?

   Answer: The Nine Chapters

   Why: Chapter Eight, Rectangular Arrays, works through 18 such problems with two to five equations each.

4. 4 Easy

   At what angle do two nonzero vectors have a dot product of 0?

   Answer: 90°

   Why: The dot product equals $|a||b|\cos\theta$, so it comes out positive below 90° and negative above 90°.

5. 5 Medium

   Which shape do all solutions of $x + y + z = 1$ form?

   Answer: Plane

   Why: With two unknowns a linear equation draws a line, and each extra unknown adds a dimension, so three unknowns give a plane.

6. 6 Hard

   How few numbers can describe any rotation in 3D?

   Answer: 3

   Why: A 3×3 rotation matrix has 9 entries, but its columns must have unit length and meet at right angles, and those 6 conditions leave 3 free.