- #1
The attached image in post 1 of this thread doesn't specify any particular matrix, so it's not possible to determine the entries of A - 2I. You might be confusing what was in the hand-drawn sketch of the previous thread from the OP, which itself was confused.fresh_42 said:$$
A-2I =\begin{pmatrix}1&-2\\1&-2\end{pmatrix}-\begin{pmatrix}2&0\\0&2\end{pmatrix}=\begin{pmatrix}-1&-2\\1&-4\end{pmatrix}
$$
and thus ##(A-2I)^{-1}=\dfrac{1}{6}\begin{pmatrix}-4&2\\-1&-1\end{pmatrix}##
fresh_42 said:So ##A-2I## is invertible. Since we have ##A\cdot \begin{pmatrix}x\\y\end{pmatrix}=2I\cdot \begin{pmatrix}x\\y\end{pmatrix},## we get
$$
A\cdot \begin{pmatrix}x\\y\end{pmatrix}-2I\cdot \begin{pmatrix}x\\y\end{pmatrix}= (A-2I)\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}
$$
Applying ##(A-2I)^{-1}## on both sides results in
$$
(A-2I)^{-1}\cdot (A-2I)\cdot \begin{pmatrix}x\\y\end{pmatrix}= I\cdot \begin{pmatrix}x\\y\end{pmatrix} =\begin{pmatrix}x\\y\end{pmatrix} =(A-2I)^{-1}\cdot \begin{pmatrix}0\\0\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}
$$
Occam's razor. Why complicate things? I do not debate typos.Mark44 said:The attached image in post 1 of this thread doesn't specify any particular matrix, so it's not possible to determine the entries of A - 2I.
The OP is already sufficiently confused as evidenced in the thread title, in thinking that finding the inverse of a matrix plays any role in finding eigenvalues. Muddying up the water by tossing in a specific matrix where none was given doesn't help alleviate that confusion.fresh_42 said:Occam's razor. Why complicate things?
The inverse method is used to find the eigenvalues of a matrix by first finding the inverse of the matrix and then using the inverse to calculate the eigenvalues.
The inverse of a matrix can be found by using the Gaussian elimination method or by using the adjugate matrix method. Both methods involve performing a series of row operations on the matrix to reduce it to its inverse form.
No, not all matrices have an inverse. A matrix must be square and have a non-zero determinant in order to have an inverse. Matrices with zero determinants are called singular matrices and do not have an inverse.
The inverse method works by first finding the inverse of the matrix, which is a new matrix that when multiplied by the original matrix results in the identity matrix. Then, the eigenvalues can be calculated by solving the characteristic equation using the inverse matrix.
Yes, there are some limitations to using the inverse method. It can only be used for square matrices with non-zero determinants. Additionally, the inverse method can be computationally expensive for large matrices, so other methods may be more efficient in those cases.