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How to use
This tool brings the core small-matrix operations together. Use 2×2 or 3×3 matrices for addition, subtraction and multiplication, or calculate determinant, transpose and inverse when mathematically defined.
- Choose 2×2 or 3×3.
- Choose the operation.
- Enter matrix A and matrix B for binary operations.
- Calculate and review the result or a validity message.
How it is calculated
Example
Let A=[[4, 7]; [2, 6]]. Its determinant is 4×6−7×2=10, so it has an inverse: A⁻¹=(1÷10)×[[6, −7]; [−2, 4]]=[[0.6, −0.7]; [−0.2, 0.4]]. Multiplying A by this inverse returns the identity matrix.
Important notes
An inverse exists only for a square nonsingular matrix. Matrix multiplication is generally not commutative: A×B may differ from B×A.
Worked examples and interpreting results
For the matrix with rows (1, 2) and (3, 4), the determinant is −2. For the identity matrix with diagonal (1, 1) and zeros elsewhere, it is 1.
| Case | Calculation | Result |
|---|---|---|
| First determinant | 1 × 4 − 2 × 3 | -2 |
| Identity determinant | 1 × 1 − 0 × 0 | 1 |
How to check the result
Select size 2 and determinant, then enter elements row by row. Matrix B is unused for this operation. A nonzero determinant permits a theoretical inverse.
Frequently asked questions
When does an inverse not exist?
When the determinant is zero or numerically indistinguishable from zero.
Is A×B equal to B×A?
Not in general; order matters.
Is 3×3 supported?
Yes for the operations shown in the calculator.
What does a zero determinant mean?
It means the square matrix is singular and has no ordinary inverse. It also indicates linear dependence among rows or columns, so an associated system does not have a unique solution.