Editing Adjugate matrix (section)

=== Column substitution and Cramer's rule ===
{{see also|Cramer's rule}}

Partition {{math|'''A'''}} into [[column vector]]s:
:<math>\mathbf{A} = \begin{bmatrix}\mathbf{a}_1 & \cdots & \mathbf{a}_n\end{bmatrix}.</math>
Let {{math|'''b'''}} be a column vector of size {{math|''n''}}.  Fix {{math|1 ≤ ''i'' ≤ ''n''}} and consider the matrix formed by replacing column {{math|''i''}} of {{math|'''A'''}} by {{math|'''b'''}}:
:<math>(\mathbf{A} \stackrel{i}{\leftarrow} \mathbf{b})\ \stackrel{\text{def}}{=}\ \begin{bmatrix} \mathbf{a}_1 & \cdots & \mathbf{a}_{i-1} & \mathbf{b} & \mathbf{a}_{i+1} & \cdots & \mathbf{a}_n \end{bmatrix}.</math>
Laplace expand the determinant of this matrix along column {{mvar|i}}.  The result is entry {{mvar|i}} of the product {{math|adj('''A''')'''b'''}}.  Collecting these determinants for the different possible {{mvar|i}} yields an equality of column vectors
:<math>\left(\det(\mathbf{A} \stackrel{i}{\leftarrow} \mathbf{b})\right)_{i=1}^n = \operatorname{adj}(\mathbf{A})\mathbf{b}.</math>

This formula has the following concrete consequence.  Consider the [[linear system of equations]]
:<math>\mathbf{A}\mathbf{x} = \mathbf{b}.</math>
Assume that {{math|'''A'''}} is [[singular matrix|non-singular]].  Multiplying this system on the left by {{math|adj('''A''')}} and dividing by the determinant yields
:<math>\mathbf{x} = \frac{\operatorname{adj}(\mathbf{A})\mathbf{b}}{\det \mathbf{A}}.</math>
Applying the previous formula to this situation yields '''Cramer's rule''',
:<math>x_i = \frac{\det(\mathbf{A} \stackrel{i}{\leftarrow} \mathbf{b})}{\det \mathbf{A}},</math>
where {{math|''x''<sub>''i''</sub>}} is the {{mvar|i}}th entry of {{math|'''x'''}}.