Fill in a matrix up to 6×6 to get its determinant, inverse, RREF, rank, characteristic polynomial or eigenvalues, or add, subtract and multiply two matrices – with cofactor expansion, Gauss–Jordan elimination and each row-by-column sum written out.
What each operation shows
| Operation | Working shown | Needs |
|---|---|---|
| Determinant | ad - bc for 2×2; cofactor expansion along the row or column with the most zeros for 3×3; row reduction to triangular form for larger | Square matrix |
| Inverse | The 2×2 formula, or Gauss–Jordan elimination on [A \mid I], then a check that AA^{-1} = I | Square, determinant ≠ 0 |
| Multiply | Each entry as row × column: products, then sums | Columns of A = rows of B |
| Add / subtract | Entry by entry | Same size |
| Power | Repeated multiplication A^2, A^3, \dots | Square matrix |
| RREF and rank | Row operations column by column: pivot to 1, zeros above and below | Any size |
| Eigenvalues | Characteristic polynomial \det(A - \lambda I), its roots, and an eigenvector for each rational eigenvalue | Square, up to 4×4 |
Determinant of a 2×2 and a 3×3 matrix
\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bcFor a 3×3 matrix, expand along a row: each entry times the determinant of the 2×2 matrix left when its row and column are crossed out, with alternating signs +\,-\,+.
Example: \det\begin{bmatrix} 2 & 0 & 1 \\ 1 & 3 & 2 \\ 1 & 1 & 1 \end{bmatrix} along row 1
2(3 \cdot 1 - 2 \cdot 1) - 0 + 1(1 \cdot 1 - 3 \cdot 1) = 2 \cdot 1 + 1 \cdot (-2) = 0The determinant is 0, so this matrix has no inverse.
Inverse of a matrix
For a 2×2 matrix swap the diagonal entries, change the signs of the other two, and divide by the determinant:
\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}For larger matrices, write the identity matrix next to A and use row operations until the left half is the identity; the right half is then A^{-1}. If a row of zeros appears on the left, the matrix is singular and has no inverse.
Matrix multiplication
The entry in row i, column j of AB is row i of A times column j of B: multiply matching entries and add. An m × n matrix times an n × p matrix gives an m × p matrix. Order matters – AB and BA are usually different.
To solve a system of linear equations directly, use the system of equations calculator; for lengths, angles and cross products, the vector calculator.
Frequently asked questions
Why can’t I multiply these two matrices?
AB exists only when the number of columns of A equals the number of rows of B. A 2×3 matrix can multiply a 3×4 matrix (giving 2×4), but not a 2×3 matrix.
Is AB the same as BA?
Usually not – matrix multiplication is not commutative. Try A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]]: AB − BA is not the zero matrix.
How are eigenvalues found?
Solve det(A − λI) = 0. The determinant is a polynomial in λ (the characteristic polynomial); its roots are the eigenvalues. For each one, the solutions of (A − λI)v = 0 are its eigenvectors.
Can I enter fractions?
Yes. Type 1/2, −3/4 or 0.25 in a cell. All the working uses exact fractions, so answers like 3/5 stay exact.