Matrix Calculator
Calculate the determinant, inverse, and matrix multiplication for 2×2 and 3×3 matrices. Includes trace and step-by-step results.
Determinant
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Result Matrix —
Trace (sum of diagonal) —
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Determinant
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Trace —
Invertible? —
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Scalar Properties
Determinant —
Trace —
Rank —
Invertible? —
Frobenius Norm —
Inverse & Eigenvalues
Inverse Matrix —
Eigenvalues (2×2) —
How to Use This Calculator
Select Matrix Size (2×2 or 3×3) and Operation (determinant, inverse, or multiply). Enter the matrix A values (and matrix B for multiplication). Results appear instantly.
Formula
2×2 det = ad−bc • Inverse = (1/det)×adj(A) • (A×B)ᵢⱼ = Σ Aᵢₖ×Bₖⱼ
Example
[[1,2],[3,4]]: det=−2, Inverse=[[−2,1],[1.5,−0.5]], Trace=5
Frequently Asked Questions
- The determinant is a scalar value computed from a square matrix that encodes important properties of that matrix. For a 2×2 matrix [[a, b], [c, d]], det = ad − bc. For example, [[3, 2], [1, 4]]: det = 3×4 − 2×1 = 12 − 2 = 10. For a 3×3 matrix, the determinant is found by cofactor expansion along any row or column. Geometrically, the absolute value of the determinant gives the area scaling factor of the linear transformation the matrix represents.
- A determinant of zero means the matrix is singular — it has no inverse. Geometrically, a singular matrix collapses the space to a lower dimension (e.g., a 2D transformation squashes the plane to a line). Algebraically, the rows (or columns) of a singular matrix are linearly dependent — one row is a linear combination of the others. For example, [[2, 4], [1, 2]] has det = 2×2 − 4×1 = 0 because the second row is exactly half the first row.
- For a 2×2 matrix A = [[a, b], [c, d]] with determinant det = ad − bc ≠ 0, the inverse is: A⁻¹ = (1/det) × [[d, −b], [−c, a]]. Example: A = [[3, 2], [1, 4]], det = 10. A⁻¹ = (1/10) × [[4, −2], [−1, 3]] = [[0.4, −0.2], [−0.1, 0.3]]. Verify: A × A⁻¹ should equal the identity matrix I = [[1, 0], [0, 1]]. If the determinant is 0, the inverse does not exist.
- To multiply two 2×2 matrices A and B to get C = A × B, each element cᵢⱼ is the dot product of row i of A with column j of B: cᵢⱼ = Σₖ aᵢₖ × bₖⱼ. Example: A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]. Then c₁₁ = 1×5 + 2×7 = 19, c₁₂ = 1×6 + 2×8 = 22, c₂₁ = 3×5 + 4×7 = 43, c₂₂ = 3×6 + 4×8 = 50. Important: matrix multiplication is not commutative in general — A×B ≠ B×A.
- The trace of a square matrix is the sum of all elements on its main diagonal (top-left to bottom-right). For a 2×2 matrix [[a, b], [c, d]], trace = a + d. For [[1, 5], [−2, 4]], trace = 1 + 4 = 5. The trace has an important property: it equals the sum of the eigenvalues of the matrix. Also, trace(A×B) = trace(B×A), making it useful in certain proofs. The trace is invariant under similarity transformations: trace(P⁻¹AP) = trace(A).