What is the adjoint of the matrix \(\left( {\begin{array}{*{20}{c}} {\cos \left( { - \theta } \right)}&{ - \sin \left( { - \theta } \right)}\\ { - \sin \left( { - \theta } \right)}&{\cos \left( { - \theta } \right)} \end{array}} \right)\) ?
Finding the Adjoint of a Matrix
The question asks for the adjoint of the given matrix. The matrix is defined using trigonometric functions of \(-\theta\).
The given matrix is:
\[A = \left( {\begin{array}{*{20}{c}} {\cos \left( { - \theta } \right)}&{ - \sin \left( { - \theta } \right)}\\ { - \sin \left( { - \theta } \right)}&{\cos \left( { - \theta } \right)} \end{array}} \right)\]
Simplifying the Matrix Elements
We can simplify the elements of the matrix using basic trigonometric identities:
- \( \cos(-\theta) = \cos\theta \)
- \( \sin(-\theta) = -\sin\theta \)
Applying these identities to the matrix A:
The element in the first row, first column is \( \cos(-\theta) = \cos\theta \).
The element in the first row, second column is \( -\sin(-\theta) = -(-\sin\theta) = \sin\theta \).
The element in the second row, first column is \( -\sin(-\theta) = -(-\sin\theta) = \sin\theta \).
The element in the second row, second column is \( \cos(-\theta) = \cos\theta \).
So the simplified matrix A is:
\[A = \left( {\begin{array}{*{20}{c}} {\cos \theta}&{\sin \theta}\\ {\sin \theta}&{\cos \theta} \end{array}} \right)\]
Calculating the Adjoint of a 2x2 Matrix
For a general 2x2 matrix \( M = \left( {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right) \), the adjoint of the matrix, denoted as \( \text{adj}(M) \), is found by swapping the elements on the main diagonal and negating the elements on the off-diagonal. The formula is:
\[ \text{adj}(M) = \left( {\begin{array}{*{20}{c}} d&{-b}\\ {-c}&a \end{array}} \right) \]
Applying the Adjoint Formula to Matrix A
In our simplified matrix \( A = \left( {\begin{array}{*{20}{c}} {\cos \theta}&{\sin \theta}\\ {\sin \theta}&{\cos \theta} \end{array}} \right) \), we have:
- \( a = \cos\theta \)
- \( b = \sin\theta \)
- \( c = \sin\theta \)
- \( d = \cos\theta \)
Using the adjoint formula for a 2x2 matrix:
\[ \text{adj}(A) = \left( {\begin{array}{*{20}{c}} d&{-b}\\ {-c}&a \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\cos \theta}&{-(\sin \theta)}\\ {-(\sin \theta)}&{\cos \theta} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\cos \theta}&{-\sin \theta}\\ {-\sin \theta}&{\cos \theta} \end{array}} \right) \]
Final Result for the Adjoint Matrix
The adjoint of the given matrix \( \left( {\begin{array}{*{20}{c}} {\cos \left( { - \theta } \right)}&{ - \sin \left( { - \theta } \right)}\\ { - \sin \left( { - \theta } \right)}&{\cos \left( { - \theta } \right)} \end{array}} \right) \) is \( \left( {\begin{array}{*{20}{c}} {\cos \theta}&{-\sin \theta}\\ {-\sin \theta}&{\cos \theta} \end{array}} \right) \).
| Original Matrix (Simplified) | Adjoint Matrix |
|---|---|
| \( \left( {\begin{array}{*{20}{c}} {\cos \theta}&{\sin \theta}\\ {\sin \theta}&{\cos \theta} \end{array}} \right) \) | \( \left( {\begin{array}{*{20}{c}} {\cos \theta}&{-\sin \theta}\\ {-\sin \theta}&{\cos \theta} \end{array}} \right) \) |
Revision Table: Matrix Adjoint Concepts
| Concept | Description |
|---|---|
| Adjoint Matrix | The transpose of the cofactor matrix of a given matrix. |
| Cofactor | \( C_{ij} = (-1)^{i+j} M_{ij} \), where \( M_{ij} \) is the minor (determinant of the submatrix) formed by removing the \( i \)-th row and \( j \)-th column. |
| Adjoint of 2x2 Matrix | For \( \left( {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right) \), the adjoint is \( \left( {\begin{array}{*{20}{c}} d&{-b}\\ {-c}&a \end{array}} \right) \). |
| Trigonometric Identity: \( \cos(-\theta) \) | \( \cos(-\theta) = \cos\theta \) (Cosine is an even function). |
| Trigonometric Identity: \( \sin(-\theta) \) | \( \sin(-\theta) = -\sin\theta \) (Sine is an odd function). |
Additional Information: Properties of Matrix Adjoint
The adjoint of a matrix has several useful properties in linear algebra:
- For a square matrix \( A \), \( A (\text{adj}(A)) = (\text{adj}(A)) A = \det(A) I \), where \( \det(A) \) is the determinant of A and \( I \) is the identity matrix.
- If A is invertible, its inverse is given by \( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \).
- \( \text{adj}(AB) = \text{adj}(B) \text{adj}(A) \).
- \( \text{adj}(A^T) = (\text{adj}(A))^T \).
- \( \det(\text{adj}(A)) = (\det(A))^{n-1} \), where n is the order of the matrix.
Understanding the adjoint is crucial for solving systems of linear equations using Cramer's rule and finding the inverse of a matrix.
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