2×2 Matrix Calculator
Add, subtract or multiply two 2×2 matrices, or find the inverse of A. Enter each cell separately and follow row-by-column arithmetic in the result table.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Add or subtract matching cells. Matrix multiplication takes the dot product of each row of A with each column of B. Inversion uses the determinant ad−bc and the adjugate matrix.
(AB)ᵢⱼ=ΣAᵢₖBₖⱼ; det(A)=ad−bc; A⁻¹=[d,−b;−c,a]/det(A)
Worked example
A=[1,2;3,4] and B=[5,6;7,8] give AB=[19,22;43,50]. det(A)=−2 and A⁻¹=[−2,1;1.5,−0.5]. Multiplication order matters.
Assumptions and input details
Rows are entered left to right, top to bottom. Only 2×2 real numeric matrices are supported, with each input magnitude at most 1e12. B is unused and hidden in inverse mode. An inverse is rejected if the determinant after scaling by the largest cell is below 1e−14 in magnitude, including singular matrices. Other results use floating-point arithmetic, not exact symbolic algebra. Matrix multiplication is generally not commutative.
References: OpenStax: matrix inverses
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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