Free Matrix Calculator – Inverse and Determinant with Every Step Shown

The Ten Matrix Operations and How the Steps Are Produced

A matrix is a rectangular grid of numbers, and matrix arithmetic is the language of linear systems, computer graphics transforms, Markov chains and least-squares fitting. This calculator handles ten operations — addition, subtraction, scalar multiplication, matrix multiplication, transpose, determinant, inverse, rank, trace and integer powers — for sizes up to 4×4, with determinants, transposes, ranks and traces extending to 5×5. What separates it from a plain answer box is that it shows the working: the determinant is expanded term by term along the first row, and the inverse is produced by Gauss-Jordan elimination with every single row operation printed and the augmented matrix redrawn after each one. Students are usually graded on the method rather than the answer, so the method is the product. The determinant uses cofactor expansion, also called Laplace expansion by minors, as described on Wolfram MathWorld. Along the first row, det(A) = Σⱼ (−1)^(1+j) · a₁ⱼ · M₁ⱼ, where a₁ⱼ is the entry in row 1 column j and M₁ⱼ is the minor — the determinant of what remains after deleting row 1 and column j. The signs alternate, and for a 2×2 this collapses to ad − bc. The inverse uses Gauss-Jordan elimination on the augmented matrix [A | I]: the row operations that reduce A to the identity simultaneously turn I into A⁻¹. Those operations are swapping two rows, dividing a row by a non-zero constant, and subtracting a multiple of one row from another. This tool applies partial pivoting, swapping the largest pivot onto the diagonal first, which keeps the arithmetic stable. Rank counts non-zero pivot rows, and multiplication uses (AB)ᵢⱼ = Σₖ Aᵢₖ · Bₖⱼ. Take A = [[3, 0, 2], [2, 0, −2], [0, 1, 1]], the worked example Math is Fun publishes for both methods. Expanding along the top row, the minors are 2, 2 and 2, so det = 3 × 2 − 0 × 2 + 2 × 2 = 10. Since 10 is non-zero, the inverse exists. Running Gauss-Jordan on [A | I] takes six row operations — divide row 1 by 3, clear row 2, swap rows 2 and 3, scale the new pivot, then clear the last column twice — and the right half becomes [[0.2, 0.2, 0], [−0.2, 0.3, 1], [0.2, −0.3, 0]], matching the published result. Switch on exact fractions and the same inverse reads [[1/5, 1/5, 0], [−1/5, 3/10, 1], [1/5, −3/10, 0]], which is what an answer key shows. Multiplying A by that inverse returns the identity, and the tool displays that check so the answer verifies itself. The two failure modes are where a bad calculator does real damage, so both get a plain explanation. If the determinant is zero the matrix is singular and no inverse exists — [[1, 2, 3], [4, 5, 6], [7, 8, 9]] is the classic case, since its first and third columns sum to twice the second. The tool says the matrix has no inverse because its determinant is 0, rather than dividing through and printing a grid of Infinity or NaN. If you multiply matrices whose shapes do not line up, it states the rule with your actual numbers: the columns of A must equal the rows of B, because each result entry is a dot product of one row of A with one column of B. When B × A would be defined instead, it says so. Addition needs identical dimensions, and determinant, trace, inverse and power all need a square matrix. Three practical notes. Exact fractions matter more than they sound: an inverse full of 0.333333 cannot be checked against an answer key that says 1/3, so the fraction toggle converts entries to exact rationals and marks anything genuinely irrational with an approximation sign rather than faking a fraction. The row-operation sequence you see may differ from your textbook's, because partial pivoting swaps rows your lecturer might not — many valid paths reach the same inverse, and Math is Fun itself notes there is no single right order. The final matrix is what must agree. Matrix powers also grow fast, so the exponent is capped at ten. Every row operation runs in your browser, and no matrix is uploaded.

Matrix Calculator Formula & Method

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