Enter a 2×2 or 3×3 matrix and select an operation: determinant, inverse, transpose, or arithmetic with a second matrix. Results include step-by-step derivation.
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 & 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.
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
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 result
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
What is a matrix determinant?
A scalar value derived from a square matrix. For 2×2: det = ad − bc. A non-zero determinant means the matrix is invertible.
What does it mean for a matrix to be singular?
A singular matrix has determinant = 0 and no inverse. It cannot be used to solve a unique system of linear equations.
How do I multiply two matrices?
The (i, j) entry of AB is the dot product of row i of A with column j of B. Requires the number of columns of A to equal the rows of B.
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