Matrix Determinant
det(A) is a scalar: for 2×2, det=ad−bc. Zero determinant means singular (no inverse). Determinant = scale factor for area/volume under the transformation. Sarrus rule for 3×3.
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det(AB)=det(A)·det(B). det(Aᵀ)=det(A). Eigenvalues: roots of det(A−λI)=0. Geometric: |det(A)| = area/volume scale factor.
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Why: Determinant tests invertibility (det≠0), gives area/volume scaling, and appears in eigenvalue formulas (det(A−λI)=0). Cramer's rule uses determinants to solve systems.
How: 2×2: det([[a,b],[c,d]])=ad−bc. 3×3 Sarrus: add forward diagonals, subtract backward. Cofactor: expand along row/column. det(A)=0 ⟺ A singular (no inverse).
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Diagonal Products (Sarrus)
Positive vs Negative Contributions
Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
2×2 det = area of parallelogram from column vectors.
— Geometry
Cramer: xᵢ=det(Aᵢ)/det(A) where Aᵢ has b in column i.
— Cramer
Geometric Interpretation
- • 2×2: |det(A)| = area of parallelogram spanned by column vectors. Sign indicates orientation.
- • 3×3: |det(A)| = volume of parallelepiped spanned by column vectors.
- • Linear transformation: det(A) is the scaling factor for area/volume under the map x → Ax.
- • Singular matrix: det = 0 means columns are linearly dependent — transformation collapses to lower dimension.
Cramer's Rule
For Ax = b with A invertible (det(A) ≠ 0), the unique solution is xᵢ = det(Aᵢ) / det(A), where Aᵢ is A with column i replaced by b.
x₁ = det(A₁)/det(A), x₂ = det(A₂)/det(A), ...
Row Operations & Determinant
• Swap two rows: det changes sign (multiply by −1)
• Multiply row by k: det multiplies by k
• Add multiple of one row to another: det unchanged
• det(AB) = det(A) × det(B): Determinant of product is product of determinants
Formulas
| Size | Formula |
|---|---|
| 2×2 | det = ad − bc |
| 3×3 | Sarrus: sum of forward diagonals − sum of backward diagonals. Or cofactor expansion along any row/column. |
Did You Know?
Expert Tips
Choose the best row/column
For cofactor expansion, pick a row or column with the most zeros to minimize computation.
Row operations first
Use row operations to get zeros before expanding — det changes predictably (swap = −1, scale = k).
Check singularity early
If two rows or columns are proportional, det = 0. No need to compute fully.
det(AB) = det(A)det(B)
Use this to simplify: factor a matrix into easier pieces, multiply their determinants.
Frequently Asked Questions
What does a zero determinant mean?
det(A) = 0 means the matrix is singular — no inverse exists. The columns (and rows) are linearly dependent.
What is the geometric meaning of the determinant?
For 2×2: |det| = area of parallelogram spanned by columns. For 3×3: |det| = volume of parallelepiped.
What is Cramer's rule?
For Ax = b with det(A) ≠ 0, xᵢ = det(Aᵢ)/det(A) where Aᵢ is A with column i replaced by b.
What is the Sarrus rule?
For 3×3: add products of forward diagonals, subtract products of backward diagonals. Only works for 3×3.
What is the cofactor matrix?
Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ where Mᵢⱼ is the determinant of the submatrix with row i and column j removed.
What is the adjugate?
adj(A) = transpose of the cofactor matrix. Used in A⁻¹ = adj(A) / det(A).
Quick Reference
When Determinants Matter
Determinants appear in solving linear systems (Cramer's rule), finding eigenvalues (det(A−λI)=0), computing inverses (A⁻¹ = adj(A)/det(A)), testing invertibility, and measuring how linear transformations scale area or volume.
Common Determinants
| Matrix Type | Determinant |
|---|---|
| Identity Iₙ | det(I) = 1 |
| Diagonal | det = product of diagonal entries |
| Triangular | det = product of diagonal entries |
| 2×2 rotation (θ) | det = 1 (area-preserving) |
| Scaling by k | det = kⁿ for n×n matrix |
Singular matrices: A matrix with det = 0 has no inverse. The columns (and rows) are linearly dependent. Such matrices arise in systems with no unique solution or infinitely many solutions.
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