Matrix Calculator

How to use

• Pick the number of rows and columns (1 to 5) for matrices A and B and type the entries.
• A + B, A − B and A × B use both matrices; kA, Aᵀ, det(A), A⁻¹, rank(A) and Aⁿ work on one matrix. B has the same buttons, and the chosen operation recalculates whenever you change a value.
• The scalar k is used for kA and kB, and the whole-number exponent n for Aⁿ and Bⁿ.
• “Use result as A” copies the result into A so you can keep calculating, and “Swap A and B” exchanges the two matrices.

Matrix multiplication and basic operations

Below, [a b; c d] is the 2×2 matrix with first row a, b and second row c, d. Matrix multiplication follows the row-times-column rule: entry (i, j) of A × B is the sum of the products of row i of A and column j of B, taken entry by entry.

• Size rule: an m×n matrix times an n×p matrix gives an m×p matrix. The number of columns of the first must equal the number of rows of the second.
• Example: [1 2 3; 4 5 6] (2×3) × [7 8; 9 10; 11 12] (3×2) = [1·7+2·9+3·11 1·8+2·10+3·12; 4·7+5·9+6·11 4·8+5·10+6·12] = [58 64; 139 154] (2×2)
• Addition and subtraction work entry by entry, so both matrices must be the same size. kA multiplies every entry by k, and the transpose Aᵀ swaps rows and columns, turning a 2×3 matrix into a 3×2 one.
• Order matters: A × B and B × A are usually different, and depending on the sizes B × A may not exist at all.

Determinant, inverse and rank by Gaussian elimination

For a 2×2 matrix, det [a b; c d] = ad − bc, and when ad − bc ≠ 0 the inverse is [a b; c d]⁻¹ = 1/(ad − bc) × [d −b; −c a].

Larger matrices are handled with Gaussian elimination with partial pivoting. In each column the entry with the largest absolute value becomes the pivot (swapping rows if needed), and multiples of the pivot row are subtracted from the rows below to create zeros. The determinant is the product of the pivots, with the sign flipped for each row swap.

Example: eliminating [2 1 1; 4 3 3; 8 7 9] by hand without swaps gives R2 − 2R1 = [0 1 1], R3 − 4R1 = [0 3 5] and R3 − 3R2 = [0 0 2], so det = 2 × 1 × 2 = 4. The calculator swaps rows to use 8 as the first pivot, but the determinant is the same.

The inverse comes from applying the same elimination to [A | I] until the left side becomes the identity matrix I; the right side is then A⁻¹ (Gauss–Jordan elimination). The rank is the number of pivots left after elimination, which is the number of linearly independent rows. A pivot smaller than 10⁻¹² times the largest entry counts as 0.

Step-by-step worked example

Using the matrices the page opens with, A = [4 7; 2 6] and B = [1 2; 3 4]:

• A × B = [4·1+7·3 4·2+7·4; 2·1+6·3 2·2+6·4] = [25 36; 20 28]
• det(A) = 4 × 6 − 7 × 2 = 24 − 14 = 10
• Inverse by formula: A⁻¹ = 1/10 × [6 −7; −2 4] = [0.6 −0.7; −0.2 0.4]
• Check by elimination: starting from [4 7 | 1 0; 2 6 | 0 1], R2 − 0.5R1 → [0 2.5 | −0.5 1], R2 ÷ 2.5 → [0 1 | −0.2 0.4], R1 − 7R2 → [4 0 | 2.4 −2.8], R1 ÷ 4 → [1 0 | 0.6 −0.7]
• Verify: A × A⁻¹ = [1 0; 0 1]. For powers, A² = A × A = [30 70; 20 50], A⁰ = I and A⁻² = (A⁻¹)² = [0.5 −0.7; −0.2 0.3].

Good to know

• The determinant, inverse and powers need a square matrix (same number of rows and columns). Transpose, scalar multiple and rank work for any size.
• A matrix with determinant 0 is singular: it has no inverse, and negative powers such as A⁻¹ or A⁻² can’t be calculated either. Example: [1 2; 2 4] has det = 1 × 4 − 2 × 2 = 0 and rank 1.
• Empty cells count as 0, and a note appears under the result when that happens. A forgotten entry changes the answer, so check it.
• Results are shown to 12 significant digits, and tiny rounding leftovers are set to 0, so A × A⁻¹ appears as an exact identity matrix.