Matrix Calculator
Matrix calculator for 2×2, 3×3, and 4×4 matrices. Add, multiply, determinant, inverse, and transpose with step-by-step working. Includes row reduction and eigenvalues.
Quick Calculator Get a fast estimate
Matrix A
Determinant of A
5
How to Use the Matrix Calculator
Enter the values in the matrix cells and select the operation: Determinant, Inverse, Transpose, Add (A+B), or Multiply (A×B). Results update as you type.
The Extended Calculator adds 3×3 support with step-by-step determinant working. The Professional Calculator includes Gaussian elimination, rank, and eigenvalues.
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Extended Calculator More options, charts, and scenario comparison
Matrix A (3×3)
Determinant
8
Step-by-step calculation
Expand along row 1:
= 2×(3×2 − 1×1) − 1×(1×2 − 1×0) + 0×(1×1 − 3×0)
= 2×5 − 1×2 + 0×1
= 10 − 2 + 0
= 8
Matrix Operations Reference
2×2 Determinant: det([[a,b],[c,d]]) = ad − bc
2×2 Inverse: A⁻¹ = (1/det) × [[d,−b],[−c,a]]
Matrix Product (A×B): C[i][j] = Σ A[i][k]×B[k][j]
Transpose: (Aᵀ)[i][j] = A[j][i]
3×3 Determinant (cofactor expansion along row 1):
det = a(ei−fh) − b(di−fg) + c(dh−eg)
Matrix Properties
| Property | Condition | Implication |
|---|---|---|
| Invertible | det ≠ 0 | Unique solution to Ax=b |
| Singular | det = 0 | No inverse, system has 0 or ∞ solutions |
| Symmetric | A = Aᵀ | Real eigenvalues |
| Identity | diagonal 1s, rest 0 | A × I = A |
| Full rank | rank = n | Matrix is invertible |
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Professional Calculator Complete parameters, sensitivity analysis, and detailed breakdown
Matrix A
Reduced Row Echelon Form (RREF)
1
0
0
0
1
0
0
0
1
Rank: 3
Row operations (9 steps)
- R1 ÷ 2
- R2 − -3×R1
- R3 − -2×R1
- R2 ÷ 0.5
- R1 − 0.5×R2
- R3 − 2×R2
- R3 ÷ -1
- R1 − -1×R3
- R2 − 1×R3
Frequently Asked Questions
For matrix [[a,b],[c,d]], determinant = ad − bc. Example: [[3,2],[1,4]] → 3×4 − 2×1 = 12 − 2 = 10.
Zero determinant means the matrix is singular: no inverse exists and the rows/columns are linearly dependent. A system Ax=b will have no unique solution.
Swap the diagonal elements (a↔d), negate the off-diagonal (b and c become −b and −c), then divide every element by the determinant. The result satisfies A × A⁻¹ = I.
Transpose swaps rows and columns. Element at row i, column j moves to row j, column i. For a symmetric matrix, the transpose equals the original matrix.
Rank is the number of linearly independent rows, found via row reduction. For a 3×3 matrix: rank 3 = invertible, rank 2 = one dependent row, rank 1 = all rows proportional.