Understanding Upper Triangular Matrices
An upper triangular matrix A has the form:
- a11 a12 a13
- 0 a22 a23
- 0 0 a33
Given that the eigenvalues are 1, 2, and -3, the diagonal entries (a11, a22, a33) of A are these eigenvalues.
Finding the Inverse of A
The inverse of an upper triangular matrix is also upper triangular. For a 3x3 upper triangular matrix, the inverse can be expressed using the matrix itself and its powers.
Expressing the Inverse in Terms of A, I, and A²
The inverse of A can be derived using the relationship of eigenvalues:
- Since the eigenvalues of A are λ1, λ2, and λ3, the eigenvalues of the inverse matrix A^(-1) will be 1/λ1, 1/λ2, and 1/λ3.
- Therefore, the eigenvalues of A^(-1) are 1, 1/2, and -1/3.
To express A^(-1) in terms of A, I, and A², we can use the following relationship:
Using Cayley-Hamilton Theorem
According to the Cayley-Hamilton theorem, a matrix satisfies its characteristic polynomial:
- p(λ) = (λ - 1)(λ - 2)(λ + 3) = λ^3 + 4λ^2 - 6λ - 6
The matrix A satisfies:
- A^3 + 4A² - 6A - 6I = 0
From this, we can isolate A^(-1):
- A^(-1) = (1/6)(A² + 4A + 6I)
Conclusion
Thus, the inverse of matrix A can be expressed as:
- A^(-1) = (1/6)(A² + 4A + 6I)
This representation allows for efficient computation of the inverse using the original matrix and its square.