Method
- The adjoint of a square matrix is important , since it enables us to form the inverse of the matrix.
- If each element of the adjoint of A is divided by the value of the determinant of A, i.e |A| , (provided |A| is not equal to 0),the resulting matrix is called the inverse of A and is denoted by A-1:
The formula for the inverse of a matrix is:
\[ \textcolor{red}{A^{-1} = \frac{1}{\det A} \cdot \operatorname{adj} A} \]
To form the inverse of a square matrix A:
(a) Evaluate the determinant of A, i.e. |A|
(b) Form a matrix C of the cofactors of the elements of |A|
(c) Write the transpose of C, i.e. CT, to obtain the adjoint of A
(d) Divide each element of CT by |A|
(e) The resulting matrix is the inverse A-1 of the original matrix A.
Example
\[\Large
A = \begin{bmatrix}
2 & 3 & 5 \\
4 & 1 & 6 \\
1 & 5 & 0
\end{bmatrix}
\]
\[\Large C = \begin{bmatrix} -24 & 6 & 15 \\ 20 & -5 & -5 \\ 13 & 8 & -10 \end{bmatrix} \]
\[\Large C^T = \begin{bmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{bmatrix} = adj A \]
\[\Large A^{-1} = \begin{bmatrix} -24/45 & 20/45 & 13/45 \\ 6/45 & -5/45 & 8/45 \\ 15/45 & -5/45 & -10/45 \end{bmatrix} = 1/45 = \begin{bmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{bmatrix} \]
\[\Large C = \begin{bmatrix} -24 & 6 & 15 \\ 20 & -5 & -5 \\ 13 & 8 & -10 \end{bmatrix} \]
\[\Large C^T = \begin{bmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{bmatrix} = adj A \]
\[\Large A^{-1} = \begin{bmatrix} -24/45 & 20/45 & 13/45 \\ 6/45 & -5/45 & 8/45 \\ 15/45 & -5/45 & -10/45 \end{bmatrix} = 1/45 = \begin{bmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{bmatrix} \]
Select the True statement