How to use this calculator
- Enter or choose operation.
- Enter or choose matrix A.
- Enter or choose matrix B.
- The result (exact) updates as you type. Use Copy result or Share to keep it.
Method
det and rank by Gaussian elimination; A⁻¹ by Gauss–Jordan elimination on [A | I]; (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ
- A, B
- the matrices
- I
- the identity matrix
- i, j
- row and column positions
Worked example
- det A = 2 × 4 − 1 × 7 = 1.
- For a 2 × 2 matrix, swap the diagonal, negate the off-diagonal and divide by det: [4, −1; −7, 2] ÷ 1.
- Check: [2 1; 7 4] × [4 −1; −7 2] = [1 0; 0 1].
The inverse of [2 1; 7 4] is [4 −1; −7 2].
What the result means
A matrix has an inverse only when its determinant is not zero. The rank is the number of independent rows; a square matrix of full rank is invertible. Results are written row by row, with rows separated by semicolons.
Assumptions and limits
- Entries are real numbers entered as decimals or fractions.
- Up to 10 × 10. Eigenvalues and decompositions are not covered.
Questions
When can I multiply two matrices?
A × B needs the number of columns of A to equal the number of rows of B. An m × n matrix times an n × p matrix gives an m × p matrix, and in general A × B ≠ B × A.
Why does it say the matrix is singular?
Its determinant is zero: some row is a combination of the others, so the matrix squashes space flat and cannot be undone.