Matrix Calculator — 2×2 Determinant, Inverse & Multiply
Quick answer: Enter a 2×2 matrix to find its determinant, trace, and inverse — or multiply two matrices.
Perform 2×2 matrix operations: find the determinant (ad − bc), trace (a + d), and inverse matrix. Also multiply two 2×2 matrices (A × B). Essential for linear algebra, physics, and computer graphics.
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Last reviewed: April 2026Report an error
Matrix A [a b; c d]
Determinant
-2
det(A) = -2. Trace = 5. Matrix is invertible
Determinant (ad - bc)
-2
Trace (a + d)
5
Invertible?
Yes
Inverse Matrix A⁻¹
-2
1
1.5
-0.5
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How to Use This Matrix Calculator Calculator
- 1Enter the four elements of Matrix A: a, b (row 1) and c, d (row 2).
- 2For multiplication, also enter Matrix B elements.
- 3Select "Matrix A Operations" for det/inverse/trace, or "A × B Multiply" for the product.
- 4Read the results.
Frequently Asked Questions
- For a 2×2 matrix [a b; c d], the determinant = ad − bc. It tells you whether the matrix is invertible (det ≠ 0) and represents the scaling factor of the linear transformation.
- A⁻¹ = (1/det) × [d −b; −c a]. Only exists when det ≠ 0. A × A⁻¹ = the identity matrix [1 0; 0 1].
- Each entry = dot product of a row from A and a column from B. (AB)₁₁ = a×e + b×g, (AB)₁₂ = a×f + b×h, (AB)₂₁ = c×e + d×g, (AB)₂₂ = c×f + d×h.
- No. A×B ≠ B×A in general. Matrix multiplication is non-commutative — the order matters.
- The trace is the sum of the diagonal elements: a + d. It equals the sum of the eigenvalues.
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