Cofactor Calculator (3x3)

This cofactor calculator finds the minor and the cofactor of any single entry in a 3 by 3 matrix, the two building blocks used to expand a determinant, build the adjugate and compute a matrix inverse. Enter the nine entries of your matrix into the m11 through m33 fields, then choose the row i and column j of the entry you want using the two dropdowns. The calculator deletes that row and column, takes the determinant of the remaining 2 by 2 matrix to find the minor, then multiplies it by the checkerboard sign of minus one to the power i plus j to give the cofactor. The results panel shows the cofactor as the headline figure, alongside the minor and the sign applied, so you see how the answer was built, not just the number itself. Change any entry or the chosen position and the result updates straight away, making it easy to check working by hand or step through every position in a matrix. Try the default identity matrix at row 1, column 1 for a simple worked case where the minor is 1, the sign is plus, and the cofactor is 1. It is a study and checking tool for algebra and linear algebra students, so enter accurate values and treat the output as a check on your own working, not a substitute for understanding the method.

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m11

m12

m13

m21

m22

m23

m31

m32

m33

Row i

Col j

Minor

Sign (-1)^(i+j)

The formula

The minor M_ij is the determinant of the 2 by 2 matrix left after deleting row i and column j. The cofactor is C_ij = (minus 1) to the power (i + j) times M_ij. The signs form a checkerboard pattern starting with plus at the top left.

Worked example

For the 3 by 3 identity matrix, the cofactor at position (1, 1) deletes the first row and column, leaving the 2 by 2 identity with determinant 1, and the sign is plus, so the cofactor is 1. Use the default matrix and position 1, 1.

Frequently asked questions

What is a minor?

The determinant of the smaller matrix left after deleting the row and column of a chosen entry.

What is a cofactor?

The minor multiplied by a sign, (minus 1) to the power (row plus column), giving the checkerboard pattern of signs.

What are cofactors used for?

Expanding determinants, building the adjugate, and computing the inverse of a matrix.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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