Editing Minor (linear algebra) (section)

===First minors===

If {{math|'''A'''}} is a square matrix, then the ''minor'' of the entry in the {{mvar|i}}-th row and {{mvar|j}}-th column (also called the {{math|(''i'', ''j'')}} ''minor'', or a ''first minor''<ref>Burnside, William Snow & Panton, Arthur William (1886) ''[https://books.google.com/books?id=BhgPAAAAIAAJ&pg=PA239 Theory of Equations: with an Introduction to the Theory of Binary Algebraic Form]''.</ref>) is the [[determinant]] of the [[submatrix]] formed by deleting the {{mvar|i}}-th row and {{mvar|j}}-th column. This number is often denoted {{math|''M''<sub>''i'', ''j''</sub>}}. The {{math|(''i'', ''j'')}} ''cofactor'' is obtained by multiplying the minor by {{math|(&minus;1){{sup|''i'' + ''j''}}}}.

To illustrate these definitions, consider the following {{nowrap|3&thinsp;×&thinsp;3}} matrix,

<math display=block>\begin{bmatrix}
1 & 4 & 7 \\
3 & 0 & 5 \\
-1 & 9 & 11 \\
\end{bmatrix}</math>

To compute the minor {{math|''M''<sub>2,3</sub>}} and the cofactor {{math|''C''<sub>2,3</sub>}}, we find the determinant of the above matrix with row 2 and column 3 removed.

<math display=block> M_{2,3} = \det \begin{bmatrix}
  1    & 4    & \Box \\
  \Box & \Box & \Box \\
  -1   & 9    & \Box \\
\end{bmatrix}= \det \begin{bmatrix}
  1 & 4 \\
  -1 & 9 \\
\end{bmatrix} = 9-(-4) = 13</math>

So the cofactor of the {{nowrap|(2,3)}} entry is

<math display=block>C_{2,3} = (-1)^{2+3}(M_{2,3}) = -13.</math>