Adjugate

The adjugate or classical adjoint of a matrix A is the transpose of the matrix of cofactors of A. The i, j cofactorAij of A is the scalar $\left(\vphantom{ -1}\right.$ -1$\left.\vphantom{ -1}\right)^{{i+j}}_{}$det A$\left(\vphantom{ i\vert j}\right.$i| j$\left.\vphantom{ i\vert j}\right)$, where A$\left(\vphantom{ i\vert j}\right.$i| j$\left.\vphantom{ i\vert j}\right)$ denotes the matrix that you obtain from A by removing the ith row and jth column.

$\blacktriangleright$ Matrices + Adjugate

$\left(\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right.$$\begin{array}{cc}
a & b \\
c & d
\end{array}$$\left.\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right)$, adjugate: $\left(\vphantom{
\begin{array}{rr}
d & -b \\
-c & a
\end{array}
}\right.$$\begin{array}{rr}
d & -b \\
-c & a
\end{array}$$\left.\vphantom{
\begin{array}{rr}
d & -b \\
-c & a
\end{array}
}\right)$


Note    The product of a matrix with its adjugate is diagonal, with the entries on the diagonal equal to the determinant of the matrix.

$\displaystyle \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right.$$\displaystyle \begin{array}{cc}
a & b \\
c & d
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d
\end{array}
}\right)$$\displaystyle \left(\vphantom{
\begin{array}{rr}
d & -b \\
-c & a
\end{array}
}\right.$$\displaystyle \begin{array}{rr}
d & -b \\
-c & a
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{rr}
d & -b \\
-c & a
\end{array}
}\right)$ =  $\displaystyle \left(\vphantom{
\begin{array}{cc}
ad-bc & 0 \\
0 & ad-bc
\end{array}
}\right.$$\displaystyle \begin{array}{cc}
ad-bc & 0 \\
0 & ad-bc
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
ad-bc & 0 \\
0 & ad-bc
\end{array}
}\right)$

$\blacktriangleright$ Matrices + Adjugate

$\left[\vphantom{
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}
}\right.$$\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}
}\right]$, adjugate: $\left[\vphantom{
\begin{array}{rrrr}
3384 & 469 & -3183 & 7130 \\
3329 & ...
...
4068 & -261 & 6896 & -2976 \\
275 & 840 & 85 & -1116
\end{array}
}\right.$$\begin{array}{rrrr}
3384 & 469 & -3183 & 7130 \\
3329 & 301 & -3200 & -8153 \\
4068 & -261 & 6896 & -2976 \\
275 & 840 & 85 & -1116
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
3384 & 469 & -3183 & 7130 \\
3329 & ...
...
4068 & -261 & 6896 & -2976 \\
275 & 840 & 85 & -1116
\end{array}
}\right]$

$\blacktriangleright$ Evaluate

det$\left[\vphantom{
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}
}\right.$$\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}
}\right]$ =  77531



$\left[\vphantom{
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}
}\right.$$\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\
4 & -8 & -3 & 92 \\
-3 & -6 & 7 & 6 \\
5 & -5 & 0 & -1
\end{array}
}\right]$$\left[\vphantom{
\begin{array}{rrrr}
3384 & 469 & -3183 & 7130 \\
3329 & ...
...
4068 & -261 & 6896 & -2976 \\
275 & 840 & 85 & -1116
\end{array}
}\right.$$\begin{array}{rrrr}
3384 & 469 & -3183 & 7130 \\
3329 & 301 & -3200 & -8153 \\
4068 & -261 & 6896 & -2976 \\
275 & 840 & 85 & -1116
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
3384 & 469 & -3183 & 7130 \\
3329 & ...
...
4068 & -261 & 6896 & -2976 \\
275 & 840 & 85 & -1116
\end{array}
}\right]$

  = $\left[\vphantom{
\begin{array}{cccc}
77531 & 0 & 0 & 0 \\
0 & 77531 & 0 & 0 \\
0 & 0 & 77531 & 0 \\
0 & 0 & 0 & 77531
\end{array}
}\right.$$\begin{array}{cccc}
77531 & 0 & 0 & 0 \\
0 & 77531 & 0 & 0 \\
0 & 0 & 77531 & 0 \\
0 & 0 & 0 & 77531
\end{array}$$\left.\vphantom{
\begin{array}{cccc}
77531 & 0 & 0 & 0 \\
0 & 77531 & 0 & 0 \\
0 & 0 & 77531 & 0 \\
0 & 0 & 0 & 77531
\end{array}
}\right]$

Note    The relationship just demonstrated gives a formula for the inverse of an invertible matrix A.

A-1 = $\displaystyle {\frac{{1}}{{\det A}}}$$\displaystyle \limfunc$adjugateA