When evaluating a determinant by expanding by minors using any row or column, the signs of the terms in the expansion follow an alternating checkerboard pattern. For a $$3 \times 3$$ determinant, this sign pattern begins with a positive sign in the top-left position and alternates across both rows and columns:

$$ \begin{vmatrix} + & - & + \\ - & + & - \\ + & - & + \end{vmatrix} $$

Sign Pattern for Expanding by Minors

The minor of an entry in a $$3 \times 3$$ determinant is the $$2 \times 2$$ determinant found by eliminating the row and column in the $$3 \times 3$$ determinant that contains the entry.

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Every square matrix has a specific real number associated with it. This associated value is called the determinant of the square matrix.

Determinant of a Square Matrix

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