Calculated using standard mathematical and industry formulas.
Free online Matrix Calculator calculator for academic, engineering, and statistical math.
How to use this calculator?
Select matrix size (2×2 or 3×3) and the desired operation.
Enter matrix A entries. For arithmetic operations, also enter matrix B.
Read the result: determinant value, inverse matrix, or result matrix.
Formula & calculation method
2×2 determinant and inverse formulas
det([[a,b],[c,d]]) = ad − bc · A⁻¹ = (1/det) × [[d,−b],[−c,a]]
For 3×3, expansion along the first row via cofactors. Addition/subtraction is element-wise. Multiplication follows row × column dot products.
Step-by-step calculation example
A = [[1,2],[3,4]]: det = 1×4 − 2×3 = −2. A⁻¹ = (1/−2) × [[4,−2],[−3,1]] = [[−2,1],[1.5,−0.5]].
Key insights to remember
A 2×2 matrix [a,b; c,d] has determinant ad − bc.
A matrix is invertible if and only if its determinant ≠ 0.
Matrix multiplication is not commutative: AB ≠ BA in general.
The inverse A⁻¹ satisfies A × A⁻¹ = I (identity matrix).
How to interpret your results?
Determinant
The determinant measures the scaling factor of the linear transformation. det = 0 means the transformation collapses space (singular matrix, no inverse).
Inverse
A⁻¹ = (1/det) × adj(A). Used to solve linear systems: Ax = b → x = A⁻¹b.
Frequently asked questions
How does the Matrix Calculator calculator work?
It uses standard algebraic, geometric, or statistical theorems to return exact values.
How accurate are the results?
All calculations use double-precision floating-point arithmetic for exact accuracy.
Can I copy the step-by-step work?
Yes, step-by-step solutions are formatted clearly for homework or documentation.
How are undefined operations handled?
The calculator flags mathematical errors such as division by zero or negative square roots.
Is this tool suitable for students and engineers?
Yes, it is designed for students, teachers, engineers, and scientists.
Does it support negative numbers and decimals?
Yes, all real numbers (positive, negative, fractions, and decimals) are supported.
Related calculators
Slope Calculator — Find the slope, y-intercept, line equation, angle of inclination and perpendicular slope from two coordinate points.
Quadratic Formula Calculator — Solve quadratic equations ax² + bx + c = 0 to find real or complex roots, discriminant, and parabola vertex.
Statistics Calculator — Full descriptive statistics: five-number summary, IQR, mean, variance, standard deviation, and skewness for any dataset.