Matrix Calculator

1. Choose an operation (A+B, A·B, det(A), etc.).
2. Paste or type Matrix A (and Matrix B / vector b if needed).
3. Turn on Prefer exact fractions to keep results exact.
4. Click Calculate to get the result + optional steps and mini visual.

• Addition/Subtraction: element-by-element (same dimensions).
• Multiplication: row-by-column dot products.
• RREF/Solve/Inverse: Gaussian elimination with exact fractions.
• Determinant: elimination-based pivot tracking (exact).

Addition/Subtraction (same size matrices): C[i,j] = A[i,j] ± B[i,j]

Multiplication (A is m×n, B is n×p): C[i,j] = Σ_k=1..n A[i,k] · B[k,j]

Solving A·x = b: build the augmented matrix [A | b], then use row operations to reach RREF and read x.

Inverse: build [A | I], row-reduce until the left side is I. The right side becomes A⁻¹.

Example 1: Add two 2×2 matrices

Let A = [ [1, 2], [3, 4] ] and B = [ [5, 6], [7, 8] ]. Compute A + B.

1. Add entry-by-entry: C[1,1] = 1+5 = 6, C[1,2] = 2+6 = 8.
2. Second row: C[2,1] = 3+7 = 10, C[2,2] = 4+8 = 12.
3. Result: A+B = [ [6, 8], [10, 12] ].

Example 2: Multiply a 2×3 matrix by a 3×2 matrix

Let A = [ [1,2,3], [4,5,6] ] and B = [ [7,8], [9,10], [11,12] ]. Compute A·B.

1. C[1,1] = 1·7 + 2·9 + 3·11 = 58
2. C[1,2] = 1·8 + 2·10 + 3·12 = 64
3. C[2,1] = 4·7 + 5·9 + 6·11 = 139
4. C[2,2] = 4·8 + 5·10 + 6·12 = 154
5. Result: A·B = [ [58,64], [139,154] ].

Example 3: Solve A·x = b (2×2)

Let A = [ [2,1], [5,3] ], b = [ [1], [0] ]. Solve for x.

1. Write the augmented matrix [A | b] = [ [2,1 | 1], [5,3 | 0] ].
2. Eliminate the first column in row 2 (one valid move): R2 = R2 − (5/2)·R1.
3. Row-reduce to RREF; the last column becomes the solution vector x.
4. This calculator performs those row operations exactly (fractions), then prints x.

Tip: If you want to see the exact row operations, keep Show step-by-step enabled.