Determinant Calculator

The determinant of a square matrix from 1×1 through 6×6, in exact fractions - every row operation shown, swaps and pivots spelled out.

Size of the matrix:

Matrix A

Your input

Find the determinant of $| \begin{array}{c} 1 & 2 & 2 \\ 0 & 5 & 7 \\ 1 & 1 & 1 \end{array} |$ .

Answer

The determinant is

| \begin{array}{c} 1 & 2 & 2 \\ 0 & 5 & 7 \\ 1 & 1 & 1 \end{array} | = 2

Solution

Subtract row 1 from row 3: $R_{3} = R_{3} - R_{1}$ .

$| \begin{array}{c} 1 & 2 & 2 \\ 0 & 5 & 7 \\ 1 & 1 & 1 \end{array} | = | \begin{array}{c} 1 & 2 & 2 \\ 0 & 5 & 7 \\ 0 & −1 & −1 \end{array} |$

Add row 2 multiplied by 1/5 to row 3: $R_{3} = R_{3} + \frac{1}{5} R_{2}$ .

$| \begin{array}{c} 1 & 2 & 2 \\ 0 & 5 & 7 \\ 0 & −1 & −1 \end{array} | = | \begin{array}{c} 1 & 2 & 2 \\ 0 & 5 & 7 \\ 0 & 0 & \frac{2}{5} \end{array} |$

$| \begin{array}{c} 1 & 2 & 2 \\ 0 & 5 & 7 \\ 0 & 0 & \frac{2}{5} \end{array} | = 1 ∙ 5 ∙ \frac{2}{5} = 2$

Answer

The determinant of $| \begin{array}{c} 1 & 2 & 2 \\ 0 & 5 & 7 \\ 1 & 1 & 1 \end{array} |$ is $2$ .

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How to use the determinant calculator

Pick the size n - the grid is square, from 1×1 to 6×6.
Type a number (decimals and fractions like 1/5 both work) into every cell.
Watch the write-up: each row operation appears as a sentence plus a |before| = |after| pair of determinants, swaps flip the sign, and the diagonal product closes it out.
Read the final determinant in exact fractions, or copy it.

Add a multiple of one row to another and the determinant is unchanged - that is the whole trick: eliminate down to the diagonal and multiply the pivots. The write-up never rounds, so 1/3 stays 1/3.

What is a determinant

Every square matrix has a single number - its determinant - that summarizes how it scales space. For a 2×2 matrix [[a, b], [c, d]] it is ad − bc: with [[2, 3], [1, 4]] that gives 2×4 − 3×1 = 5. For 3×3 and larger, expansion by cofactors or row elimination breaks the problem into smaller determinants until only 2×2 cases remain, which is exactly the elimination this calculator writes out step by step.

What the determinant tells you

A determinant of zero means the matrix is singular: it squashes space flat, has no inverse, and its linear system has no unique solution. Geometrically, the absolute value is the scaling factor - a determinant of 5 stretches every area fivefold - while the sign tracks orientation: negative flips the space like a mirror. Eigenvalues connect here too, since their product always equals the determinant.