Definition:Cofactor/Element

Definition

Let:

$D = \begin {vmatrix}

a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\

\vdots &  \vdots & \ddots &  \vdots \\

a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {vmatrix}$ be a determinant of order $n$.

Let $a_{r s}$ be an element of $D$.

Let $D_{r s}$ be the determinant of order $n-1$ obtained from $D$ by deleting row $r$ and column $s$.

Then the cofactor $A_{r s}$ of the element $a_{r s}$ is defined as:

$A_{r s} := \paren {-1}^{r + s} D_{r s}$

Also known as

A cofactor is also known as a signed minor.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercises -- Second Set
• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.7$ Minors and cofactors
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cofactor
• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.5$: Determinants
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cofactor
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cofactor
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): cofactor