Adjugate Matrix
The adjugate matrix, or adjoint of a square matrix A. is constituted by the transposition of A's cofactor matrix. Represented as. $$ adj(A) = cof(A)^T$$
This concept is instrumental in the calculation of the inverse of a square matrix.
It's essential to note that the adjugate is only defined for square matrices.
Calculating the Adjoint Matrix
The adjugate of a matrix can be obtained through the following procedure:
- Calculate the cofactor matrix of A, denoted as cof(A)
The cofactor of each element in A is obtained by multiplying (-1)^(i+j) with the determinant of the submatrix, which results from excluding the i-th row and j-th column. $$ cof(A) $$The cofactor of an element in a matrix is (-1)(i+j) times the determinant of the submatrix obtained by removing the i-th row and the j-th column.
In layman terms, for a 2x2 matrix $$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ the cofactor matrix is cof(A) $$ cof(A) = \begin{pmatrix} d & -c \\ -b & a \end{pmatrix} $$ For a 3x3 matrix $$ A= \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} $$ the cofactor matrix is computed as shown below. $$ cof(A)= \begin{pmatrix} \det \begin{pmatrix} e & f \\ h & i\end{pmatrix} & - \det \begin{pmatrix} d & f \\ g & i\end{pmatrix} & \det \begin{pmatrix} d & e \\ g & h \end{pmatrix} \\ - \det \begin{pmatrix} b & c \\ h & i\end{pmatrix} & \det \begin{pmatrix} a & c \\ g & i\end{pmatrix} & - \det \begin{pmatrix} a & b \\ g & h \end{pmatrix} \\ \det \begin{pmatrix} b & c \\ h & i\end{pmatrix} & - \det \begin{pmatrix} a & c \\ d & f\end{pmatrix} & \det \begin{pmatrix} a & b \\ d & e \end{pmatrix} \end{pmatrix} $$ For each element, you remove the corresponding row and column, then calculate the determinant, alternating between positive and negative signs.
- Proceed to transpose the cofactor matrix, swapping its rows with columns
This provides the adjugate of A, expressed as adj(A) $$ adj(A) = cof(A)^T $$What's the Point of the Adjoint Matrix? The adjugate plays a critical role in finding the inverse of a square matrix. For an invertible matrix A, the inverse is given by 1/det(A) multiplied by the adjugate of A, where det(A) is the determinant of A: $$ A^{-1} = \frac{1}{\det(A)} \cdot adj(A) $$
In the sections ahead, we delve into practical instances and explore the principal attributes of adjugate matrices in the field of linear algebra.
Example of adjugate matrix
Consider the 2x2 matrix
$$ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$
where a=1, b=2, c=3, and d=4.
The cofactor matrix, cof(A), becomes
$$ cof(A) = \begin{pmatrix} d & -c \\ -b & a \end{pmatrix} $$
$$ cof(A) = \begin{pmatrix} 4 & -3 \\ -2 & 1 \end{pmatrix} $$
Transposing this yields the adjugate of A
$$ adj(A) = cof(A)^T = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} $$
Let's look at another example.
Consider a 3x3 matrix
$$ A = \begin{pmatrix} 1 & 2 & 0 \\ 3 & 4 & 5 \\ 0 & 1 & 1 \end{pmatrix} $$
Calculate the cofactor matrix,
$$ cof(A)= \begin{pmatrix} \det \begin{pmatrix} 4 & 5 \\ 1 & 1 \end{pmatrix} & - \det \begin{pmatrix} 3 & 5 \\ 0 & 1 \end{pmatrix} & \det \begin{pmatrix} 3 & 4 \\ 0 & 1 \end{pmatrix} \\ - \det \begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} & \det \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} & - \det \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \\ \det \begin{pmatrix} 2 & 0 \\ 4 & 5\end{pmatrix} & - \det \begin{pmatrix} 1 & 0 \\ 3 & 5 \end{pmatrix} & \det \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \end{pmatrix} $$
$$ cof(A)= \begin{pmatrix} \det (-1) & - \det (3) & \det (3) \\ - \det (2) & \det (1) & - \det (1) \\ \det (10) & - \det (5) & \det (-2) \end{pmatrix} $$
$$ cof(A)= \begin{pmatrix} -1 & - 3 & 3 \\ -2 & 1 & -1 \\ 10 & -5 & -2 \end{pmatrix} $$
and then transpose it to get the adjoint of A as
$$ adj(A) = cof(A)^T = \begin{pmatrix} -1 & - 2 & 10 \\ -3 & 1 & -5 \\ 3 & -1 & -2 \end{pmatrix} $$
This approach applies to square matrices of larger dimensions, though the calculations become more complex with increasing matrix size.
Properties of Adjugate Matrices
Outlined below are some properties of adjugate matrices:
- Inverse Property
If A is invertible, its inverse is expressible in terms of the adjugate and the determinant of A $$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)$$ - Determinant Property
For a square matrix A $$A \cdot \text{adj}(A) = \text{det}(A) \cdot I$$ with I being an identity matrix of the same dimension as A - Transpose Property
The adjoint of the transpose of A equals the transpose of the adjoint of A $$\text{adj}(A^T) = (\text{adj}(A))^T$$ - Product Property
For two square matrices A and B of the same order: $$\text{adj}(AB) = \text{adj}(B) \cdot \text{adj}(A)$$ This property shows that the operation of adjoint behaves similarly to inversion with respect to the product of matrices. - Power Property
For a square matrix A and a non-negative integer k $$\text{adj}(A^k) = (\text{adj}(A))^k$$ - Double Adjugate Property
The adjoint of the adjoint of A equals det(A)(n-2)·A for a matrix A of order n $$\text{adj}(\text{adj}(A)) = \text{det}(A)^{n-2} \cdot A$$
These properties help to delineate the behavior of adjugate matrices in major matrix operations.
