Matrix CalculatorFree online math calculator — instant results, no sign-up
Determinant and inverse of a 2×2 or 3×3 matrix (rows on lines).
About the Matrix Calculator
This calculator computes the determinant, transpose, and invertibility status for 2×2 and 3×3 square matrices. It is designed for students, engineers, and researchers who need quick linear algebra checks without manual cofactor expansion. Entering rows line-by-line produces the determinant rounded up to six decimal places alongside the transposed matrix.
- Matrix
- A 2×2 or 3×3 grid of numbers entered with one row per line and elements separated by spaces or commas.
Formula and method
For 2×2: det(A) = a₁₁a₂₂ − a₁₂a₂₁ For 3×3: det(A) = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁) Transpose: T_ij = a_ji
- a_ijMatrix element located at row i and column j
- det(A)Determinant of matrix A
- T_ijElement of the transposed matrix at row i and column j
The tool parses the input text by row, splitting numeric values into an array. It calculates the determinant via standard 2×2 cross-multiplication or 3×3 first-row cofactor expansion, rounding the result to 6 decimal places. It swaps rows and columns to construct the transpose and flags the matrix as invertible whenever the determinant is non-zero.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Matrix2 1 1 1 3 2 1 0 0
Results
- Determinant-1
- Transpose2 1 1 | 1 3 0 | 1 2 0
- InvertibleYes
How to use the Matrix Calculator
- 1Enter each matrix row on a separate line, separating numbers with spaces.
- 2Evaluate the determinant using 2×2 cross-multiplication or 3×3 cofactor expansion along the top row.
- 3Swap row and column indices to construct the transpose, displayed with rows separated by vertical bars.
- 4Check if the determinant is non-zero to determine whether the matrix is invertible.
Reading your result
The Determinant indicates the scaling factor of the transformation; a determinant of zero signifies a singular matrix with linearly dependent rows or columns. The Transpose displays the flipped matrix where the original columns become rows. 'Invertible' shows 'Yes' if the determinant is non-zero, meaning a unique matrix inverse exists.
Limitations
The calculator only supports 2×2 and 3×3 square matrices; dimensions 4×4 and larger are not evaluated. It displays whether an inverse exists, but it does not output the explicit inverse matrix elements. Entering non-square dimensions or mismatched row lengths will lead to calculation errors.
Common questions
What makes a matrix invertible?+
A square matrix is invertible if and only if its determinant is not equal to zero. If the determinant is zero, the matrix is singular and has no multiplicative inverse.
How should I format negative numbers or decimals?+
Type standard negative signs and decimal points directly, such as '-2.5 0 1.25' on a row line.
Why does the transpose output use vertical bars?+
The vertical bar '|' separates the rows of the resulting transposed matrix, keeping each row's space-delimited elements distinct.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.