2×2 to 5×5 Matrices | Step-by-Step Solutions | Two Methods
✓ 4 Matrix Sizes - 2×2 to 5×5 support
✓ 2 Methods - Cofactor & row reduction
✓ Step-by-Step - Complete solutions
✓ Properties - Singular/invertible check
2×2 to 5×5 Matrices | Step-by-Step Solutions | Multiple Methods
Determinant: A scalar value that determines if a matrix is invertible (det ≠ 0) or singular (det = 0).
💡 Tip: Cofactor shows exact steps, Row Reduction is faster for large matrices!
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Non-Singular
det ≠ 0, invertible
Singular
det = 0, not invertible
Calculate determinants for 2×2, 3×3, 4×4, and 5×5 square matrices. Dynamic grid adapts to selected size with easy input interface.
Choose between cofactor expansion for exact calculations or row reduction for efficient computation with complete step-by-step working.
Automatically analyzes matrix properties including singular/non-singular detection, invertibility check, and rank estimation based on determinant.
A Determinant Calculator is specialized mathematical tool computing scalar value called determinant from square matrix elements fundamental concept linear algebra with profound theoretical practical implications determinant single number encoding important matrix properties invertibility linear independence transformation scaling providing crucial information about matrix behavior system solvability geometric interpretations determinant calculated from square matrices only same number rows columns represented vertical bars notation vertical bar A vertical bar or det parenthesis A parenthesis commonly det A determinant value can be positive negative zero each having specific meaning positive determinant indicates transformation preserves orientation negative indicates orientation reversal zero determinant indicates dimensional collapse singularity fundamental formulas two-by-two matrix determinant equals ad minus bc where a b c d are matrix elements calculated by multiplying main diagonal elements subtracting product anti-diagonal elements simplest determinant formula foundation understanding larger matrices three-by-three matrix determinant calculated using cofactor expansion along any row column typically first row formula a times ei minus fh minus b times di minus fg plus c times dh minus eg where letters represent matrix positions showing three 2×2 minors calculated multiplied by elements with alternating signs Sarrus rule alternative method three-by-three specifically using diagonal products not generalizing larger sizes larger matrices four-by-four five-by-five beyond determinants calculated recursively using cofactor expansion breaking down into smaller minors continuing until reaching 2×2 base cases or using row reduction transforming matrix upper triangular form where determinant equals product diagonal elements computational methods cofactor expansion Laplace expansion exact method expanding along row column calculating minors determinants applying signs summing products advantages exact calculations educational value understanding determinant structure disadvantages computationally expensive large matrices factorial complexity row reduction Gaussian elimination efficient method using elementary row operations converting upper triangular form tracking sign changes from row swaps advantages computational efficiency numerical stability practical for large matrices disadvantages potential numerical errors floating point arithmetic making determinant calculator indispensable tool supporting both methods providing flexibility choosing appropriate approach based needs.
Our Free Determinant Calculator 2025 offers comprehensive advanced features exceptional functionality supporting four matrix sizes 2×2 smallest simplest matrices used introductory linear algebra basic transformations solved using direct formula ad minus bc instant calculation understanding fundamental concepts 3×3 most common size appearing throughout mathematics physics engineering courses standard size teaching determinant concepts applications calculated using cofactor expansion three minors or Sarrus rule diagonal products 4×4 advanced matrices appearing engineering problems higher dimensional transformations computer graphics quaternion calculations requiring more complex calculations cofactor expansion four 3×3 minors or row reduction efficiency 5×5 largest size calculator supports appearing advanced engineering problems physics simulations complex systems analysis multi-dimensional problems demonstrating determinant calculation scalability practical limits computational complexity unique features two calculation methods flexibility choosing appropriate method cofactor expansion educational complete working showing minor calculations sign applications detailed steps understanding determinant structure perfect learning teaching step-by-step verification row reduction computational efficient for larger matrices showing row operations pivot selections elimination steps triangular form practical applications numerical computations step-by-step solutions comprehensive detailed working every calculation showing all intermediate steps formula applications calculations verifications educational transparent helping understand process verify results learn determinant calculation methods matrix properties analysis automatic detection classification singular matrices determinant zero indicating non-invertibility dimensional collapse non-singular matrices determinant non-zero indicating invertibility full rank invertibility verification automatic check based determinant value determining whether matrix has inverse crucial before attempting inverse calculations solving linear systems rank estimation approximation based determinant value full rank when determinant non-zero reduced rank when zero helping understand matrix structure linear independence calculation history automatic tracking last ten determinant calculations helping review compare verify results work on multiple problems efficiently example values built-in examples each matrix size demonstrating calculator functionality providing starting points learning testing quick verification mobile responsive design dynamic matrix grid adapting screen size touch-friendly inputs ensuring usability smartphones tablets desktops making accessible anywhere anytime high precision results four decimal places accuracy maintaining precision throughout calculations appropriate rounding displaying ensuring reliable results scientific engineering applications error handling robust validation input checking preventing invalid calculations division by zero detecting providing clear error messages maintaining calculator stability making most powerful comprehensive user-friendly determinant calculator available online completely free unlimited calculations no registration advertisements serving students mathematics linear algebra courses engineers requiring determinant calculations scientists analyzing data transformations computer graphics professionals working transformations teachers professors educators demonstrating determinant concepts anyone needing reliable accurate determinant calculations updated 2025 standards modern computational methods mathematical best practices.
| Feature | Cofactor Expansion | Row Reduction |
|---|---|---|
| Method | Laplace expansion along row/column | Gaussian elimination to triangular form |
| Accuracy | Exact (no rounding errors) | High precision (may have small errors) |
| Speed | Slower for large matrices | Faster for large matrices |
| Best For | Learning, 2×2, 3×3 matrices | 4×4, 5×5, computational efficiency |
| Steps Shown | Minors, cofactors, contributions | Row operations, swaps, triangular form |
| Educational Value | High - shows determinant structure | Medium - shows row operations |
For 2×2 matrix [[a,b],[c,d]]: det = ad - bc. Multiply diagonal elements (a×d), multiply anti-diagonal (b×c), subtract. Example: [[4,3],[2,1]] → (4×1)-(3×2) = 4-6 = -2. Simplest formula. det=0 means singular, det≠0 means invertible. Used in linear algebra, geometry, physics.
Cofactor expansion (Laplace): det(A) = Σ((-1)^(i+j) × a_ij × M_ij). Choose row/column, multiply each element by signed minor, sum. Example 3×3: expand row 1, calculate three 2×2 minors, apply signs (+,-,+), sum. Calculator shows: positions, signs, minors, contributions. Exact method, educational value.
Row reduction converts to upper triangular, det = product of diagonal × (-1)^swaps. Steps: eliminate below diagonal, track row swaps, multiply diagonal. Example: reduce [[2,1,3],[4,2,1],[6,3,2]] to triangular → multiply diagonal. Calculator shows: operations, swaps, triangular form, product. Efficient for large matrices.
det=0 means SINGULAR matrix: not invertible, linearly dependent rows/columns, reduced rank, no unique solution. Geometric: transformation collapses dimension, zero area/volume. Examples: [[1,2],[2,4]] proportional rows. Calculator shows 'Singular', 'Not Invertible', reduced rank. Cannot compute inverse or use Cramer's rule.
3×3 using cofactor expansion: det = a(ei-fh) - b(di-fg) + c(dh-eg) for [[a,b,c],[d,e,f],[g,h,i]]. Calculate three 2×2 minors, apply signs, sum. Example: [[2,-3,1],[2,0,-1],[1,4,5]] → 2(0+4)-(-3)(10+1)+1(8-0) = 8+33+8 = 49. Alternative: Sarrus rule. Calculator shows each minor calculation.
NON-SINGULAR: det≠0, invertible, full rank, independent rows/columns, unique solution, preserves dimension. SINGULAR: det=0, not invertible, reduced rank, dependent rows/columns, no/infinite solutions, collapses dimension. Check: calculate det, if 0→singular, if ≠0→non-singular. Calculator displays type, invertibility, rank. Critical for inverse computation.
Steps: Select size, enter elements, choose method, click Calculate. Shows: det value, properties (Singular/Non-Singular, Invertible Yes/No, rank). If det≠0: invertible, can compute inverse, unique solution. If det=0: not invertible, no inverse, no/infinite solutions. Example: [[1,2,3],[0,1,4],[5,6,0]] → det=1 → Invertible:Yes. Instant verification.
Key properties: det(I)=1, det(A^T)=det(A), det(AB)=det(A)×det(B), det(kA)=k^n×det(A), row swap→×(-1), add row multiple preserves det, zero row→det=0, proportional rows→det=0, triangular→product of diagonal. Geometric: 2D area, 3D volume. Calculator demonstrates: matrix type, invertibility check, rank estimation based on properties.
Yes! Calculator supports 4×4 and 5×5. COFACTOR: expands to 3×3 or 4×4 minors, shows each minor, displays contributions. ROW REDUCTION: converts to triangular, tracks swaps, multiplies diagonal. Use: Select 4×4/5×5, enter 16/25 elements, choose method (cofactor for learning, row reduction for speed), Calculate. Shows: complete steps, properties, invertibility. For engineering, physics, advanced linear algebra.
100% free forever! All features: 2×2 to 5×5 matrices, both methods (cofactor, row reduction), complete step-by-step solutions, matrix properties (singular/invertible/rank), calculation history (10 items), example values, high precision (4 decimals), mobile responsive. No registration, payment, ads, downloads, limits. Works any device/browser. Perfect for students, engineers, scientists, teachers. Free for everyone!
Our Free Determinant Calculator 2025 is the most advanced comprehensive determinant calculation tool supporting 2×2 3×3 4×4 5×5 square matrices providing two professional calculation methods cofactor expansion row reduction complete step-by-step solutions matrix properties analysis singular non-singular detection invertibility verification rank estimation calculation history high precision results completely free unlimited use no registration advertisements serving students engineers scientists teachers anyone needing reliable accurate determinant calculations updated 2025 standards modern computational methods mathematical best practices educational excellence.