Cofactor Calculator (3x3)
This cofactor calculator finds the minor and the cofactor of a chosen entry in a 3 by 3 matrix. Enter the nine entries and pick the row and column of the element you are interested in, and the calculator deletes that row and column, takes the determinant of the 2 by 2 that remains (the minor), and applies the checkerboard sign to get the cofactor. Minors and cofactors are the building blocks of determinant expansion, matrix inverses and the adjugate. The sign follows minus one to the power i plus j. The method, a worked example and the assumptions are below.
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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