Two linear equations
Solve equations arranged as a*x + b*y = c and d*x + e*y = f. Enter the six numbers, including minus signs, rather than pasting whole equations.
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2×2 and 3×3 determinant and inverse
Enter a 2×2 or 3×3 square matrix to calculate its determinant and inverse, when it exists.
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Determinant
10
A zero determinant has no inverse; a slash denotes a fraction.
Inverse row 13/5，-7/10
Inverse row 2-1/5，2/5
Worked steps
- det(A) = 10
Enter each row on a new line with spaces or commas between values. Only 2×2 or 3×3 square matrices with finite decimals up to three places and absolute values at most one million are accepted. The 2×2 determinant is top-left times bottom-right minus top-right times bottom-left; the 3×3 determinant expands along the first row. A zero determinant has no inverse. Results are exact integers or reduced fractions: the slash means division. Fraction input, scientific notation, matrix multiplication and larger matrices are not supported.
Method and scope
The determinant is exact; when it is zero the matrix has no inverse.
This tool handles only 2×2 or 3×3 matrices and finite decimals up to three places, not matrix multiplication or arbitrary dimensions.
References:Taiwan Ministry of Education: matrices and 2×2 inverse、LibreTexts: 2×2 matrices and inverses
Purpose
Find the determinant and, when it exists, the inverse of a two-by-two or three-by-three square matrix.
What to enter
Enter one row per line, with the same two or three values on each row. Separate entries with spaces or commas.
How it works
The determinant is exact. When it is not zero, compute the inverse from the transposed cofactors divided by that determinant.
Example
For rows 4, 7 and 2, 6, the determinant is 4×6−7×2 = 10; the inverse begins with 3/5 and −7/10.
Reading the result
A determinant of zero means no inverse exists. Otherwise, each row of the inverse is shown as reduced fractions.
Limits
Only 2×2 or 3×3 finite-decimal entries up to three decimal places and absolute value one million are accepted.