Matrix Calculator — Add, Multiply, Invert, RREF, Eigenvalues with Steps

Real-world use cases

• Verify a homework determinant before turning it in

  You expanded det of a 4×4 matrix by cofactor along the first row and got 18, but you're not sure the signs alternated right. Type the 16 entries into the calculator and pick det(A). It runs Bareiss fraction-free elimination in O(n³) with exact BigInt fractions and returns the exact integer — no rounding, no "answer is approximately 17.999999". If you got 18 too, you can submit with confidence; if not, switch the operation to "RREF (with steps)" to spot which row operation in your hand calculation introduced the error.

• Invert a coefficient matrix without losing precision

  The matrix A = [[2,1,1],[1,3,2],[1,0,0]] needs to be inverted for a stats problem. Plug it in, choose A⁻¹, and the page shows both the result (exact fractions like 0/−2, 0/2, 1, −2, 1/2, 1/2, …) and every row operation that produced it. You can read the Gauss-Jordan trace top to bottom — first the row swap to bring a non-zero pivot up, then scaling each pivot to 1, then eliminating the other entries in each pivot column — and use those same operations to check your hand-derived inverse later in the semester.

• Find eigenvalues of a 3×3 stress tensor in a physics problem

  The stress tensor σ = [[3,1,0],[1,3,0],[0,0,2]] has three principal stresses (its eigenvalues). The calculator builds the characteristic polynomial λ³ − 8λ² + 20λ − 16 via Faddeev-LeVerrier (in exact rationals), then numerically resolves the roots: λ = 2, 2, 4. You also see the principal stress directions are aligned with the third coordinate axis (because the matrix is block-diagonal). All three real, no complex residue — exactly what the physics says, no spurious imaginary parts.

• Solve a linear system that turns out to have infinite solutions

  The system x + 2y + 3z = 6, 2x + 4y + 6z = 12, 3x + 5y + 7z = 15 looks like three independent equations. Plug in A = the coefficient matrix and b = (6, 12, 15), pick "Solve A·x = b". The calculator runs RREF on [A | b], finds that rank(A) = rank([A | b]) = 2 < 3, and reports "infinitely many solutions" with a particular solution and which variable is free. This is the exact moment many students realize a 3×3 system can be under-determined — the calculator names which column is free so the parametric form falls out by hand.

• Test whether a basis is linearly independent

  You have four vectors in ℝ⁵ and want to know if they form a linearly independent set. Stack them as columns of a 5×4 matrix A, pick rank(A). If rank = 4, the vectors are independent and span a 4-dimensional subspace; if rank < 4, there's a linear relation and the calculator's RREF reveals exactly which columns are pivot columns and which are dependent. Same machinery handles "is this set a basis of ℝ⁴?" — stack them in a 4×4 matrix and check det ≠ 0 or rank = 4.

Common pitfalls

• Confusing matrix multiplication with element-wise multiplication. The "×" here means A times B in the linear-algebra sense (sum of products), and it requires cols(A) = rows(B). If you wanted Hadamard (element-wise) product, that's a different operation and is not what most courses call "matrix multiplication".

• Asking for the inverse of a non-square or singular matrix. Only square matrices have inverses in the usual sense, and even square matrices need det ≠ 0. The calculator returns "singular" rather than silently giving a meaningless result; a pseudoinverse (Moore-Penrose) would be a different request.

• Reading the eigenvalue list as ordered. Eigenvalues are a multiset — the calculator may list them in any order, and repeated eigenvalues appear with multiplicity. Pair each one with its eigenvector by checking which rows of (A − λI) collapse to zero.

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