Common Mistakes
Traps in Matrices & Determinants
5 mistake patterns students fall for. Each one shows the wrong approach vs the correct approach.
Scalar multiple of determinant
FORMULAFor n×n matrix A, |kA| = k^n|A|, not k|A|.
✗ WRONG: |2A| = 2|A| for a 3×3 matrix
✓ RIGHT: |2A| = 2³|A| = 8|A|. The scalar multiplies each of the n rows.
Determinant of sum ≠ sum of determinants
FORMULA|A + B| ≠ |A| + |B| in general. This is a very common error.
✗ WRONG: |A + B| = |A| + |B|
✓ RIGHT: No shortcut for |A+B|. You must compute A+B first, then find its determinant.
Confusing D=0 cases for systems
CASE MISSD=0 does not automatically mean infinite solutions. You need all Dᵢ=0 as well.
✗ WRONG: D=0, so the system has infinite solutions
✓ RIGHT: D=0: check D₁, D₂, D₃. If any Dᵢ ≠ 0: no solution. If all Dᵢ = 0: possibly infinite (verify with substitution).
Wrong adjoint formula for nested adj
FORMULAadj(adj(A)) = |A|^{n-2}·A, and |adj(A)| = |A|^{n-1}, not |A|^{n}.
✗ WRONG: |adj(A)| = |A|^n for 3×3
✓ RIGHT: |adj(A)| = |A|^{n-1} = |A|² for 3×3. The exponent is always (n-1).
Matrix multiplication is not commutative
CASE MISSAB ≠ BA in general. Only commutative in special cases.
✗ WRONG: Rearranging AB = C to BA = C
✓ RIGHT: Never swap order unless given that AB = BA. Use left/right multiplication by inverse carefully.