Matrix Calculator

What does the matrix calculator do?

You enter matrices A and B and the tool performs the operation you choose: addition, subtraction, multiplication, scalar multiplication, transpose, determinant, inverse, rank, Gaussian elimination (reduced row echelon form) and whole number powers. Sizes range from 1 to 5. Cells accept whole numbers, decimals or fractions such as 3/4.

Why are the results fractions?

The work is done with fractions and no rounding. In an inverse you therefore see 1/3 instead of 0.3333, and a matrix whose determinant is zero never appears invertible because of rounding. You can switch the result to decimals if you prefer.

Formulas

• Product: for C = A × B the number of columns of A must equal the number of rows of B. The entry cᵢⱼ is the sum of products of row i of A and column j of B. Order matters: A × B and B × A are usually different.
• 2×2 determinant: det = a₁₁·a₂₂ − a₁₂·a₂₁.
• Larger determinants: the matrix is reduced to upper triangular form with row operations and the determinant is the product of the diagonal. Each row swap flips the sign.
• Inverse: the augmented matrix [A | I] is reduced to [I | A⁻¹]. If the determinant is zero there is no inverse.
• Rank: the number of non-zero rows in the echelon form.

Example

For A = [1 2, 3 4] the determinant is 1·4 − 2·3 = −2. The inverse is 1/(−2) times [4 −2, −3 1], which gives [−2 1, 3/2 −1/2]. Multiplying A by its inverse returns the identity matrix.

Limits

The largest size is 5×5. Cells take numbers only, so symbolic matrices and complex numbers are not supported. Eigenvalues, eigenvectors and matrix decompositions (LU, QR) are outside the scope of this tool. Decimal inputs are converted to exact fractions as typed: 0.1 is treated as exactly 1/10.