91Ó°ÊÓ

We know that matrix A is diagonalizable, which means that there exist a matrix P such that D = P^(-1)AP is a diagonal matrix. Furthermore, the eigenvalues of A are given as 1 and -1. If λ is an eigenvalue of A, and x is its associated eigenvector, then we have: Ax = λx.

Since A is diagonalizable, we can express A as a product of three matrices: A = PDP^(-1). Here, P is the matrix formed by the eigenvectors x corresponding to the eigenvalues λ, D is the diagonal matrix with λ values on the main diagonal, and P^(-1) is the inverse of matrix P. Matrix D would then look like this: D = \( \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & -1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & -1 \\ \end{pmatrix} \)

Now, let's calculate the product of A and D: A * D = (P * D * P^(-1)) * (P * D * P^(-1)) Using the associative property of matrix multiplication, we can rewrite the expression as: A * D = P * (D * P^(-1) * P) * D * P^(-1) Since the inverse of matrix P exists, the product of P^(-1) and P is the identity matrix I: A * D = P * D * I * D * P^(-1)

Noticing that the product of D and I is D, we find: A * A = P * D * D * P^(-1) Now we calculate the product of D * D: D * D = \( \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & -1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & -1 \\ \end{pmatrix} \) * \( \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & -1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & -1 \\ \end{pmatrix} \) Since D is a diagonal matrix, we only need to multiply the elements on the main diagonal: D * D = \( \begin{pmatrix} 1^2 & 0 & \cdots & 0 \\ 0 & (-1)^2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & (-1)^2 \\ \end{pmatrix} \) = \( \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \end{pmatrix} \) = I So, A * A = P * I * P^(-1) = P * P^(-1) = A^(-1)

Thus, we have shown that if matrix A is diagonalizable with eigenvalues 1 and -1, it is its own inverse: A^(-1) = A.