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Eigenvalues are a fundamental concept in linear algebra associated with a square matrix, denoting special scalar values for which there exists a non-zero vector, called an eigenvector, that when multiplied by the matrix results in a vector that is a scalar multiple of the original eigenvector. In mathematical terms, for a matrix, A, if there exists a non-zero vector v such that A v = \( \lambda \) v, then \( \lambda \) is said to be an eigenvalue of A.

In the context of the exercise, the given matrix A has eigenvalues of ±1. This property of the eigenvalues plays a crucial role in proving that A equals its own inverse because these specific eigenvalues indicate that the operation of the matrix on its eigenvectors results in vectors that are simply scaled versions of the original, either flipped or unchanged in direction, which is a distinctive characteristic leveraged in the diagonalization process.

The inverse of a matrix, denoted as A^-1, is a matrix that, when multiplied with the original matrix A, yields the identity matrix. The identity matrix I acts as a multiplicative neutral element, having the property that A A^-1 = A^-1 A = I. Not all matrices have an inverse; for a matrix to be invertible, it must be square (same number of rows and columns) and its determinant should not be zero.

For diagonalizable matrices with eigenvalues ±1 as in our exercise, finding the inverse is straightforward because the inverse of the diagonal matrix D, which contains the eigenvalues on its main diagonal, is simply the diagonal matrix with the reciprocals of those eigenvalues on its main diagonal. In our specific case, since the reciprocal of ±1 is still ±1, the inverse of D is D itself, which simplifies the process of proving A^-1 = A.

Diagonalization is the process of transforming a square matrix into a diagonal matrix D using a similarity transformation. This is done by finding a matrix P, whose columns are the eigenvectors of A, that satisfies D = P^-1 A P. A diagonal matrix has non-zero entries only on the main diagonal, which greatly simplifies many matrix operations. Not all matrices can be diagonalized, but those that can, often referred to as diagonalizable matrices, allow for more efficient computations, especially for raising matrices to powers.

In our exercise, the diagonal matrix D already has ±1 as its diagonal entries, representing the eigenvalues of A. Because of the special values of these eigenvalues, the process of diagonalization directly leads us to conclude that in this case, the matrix A indeed equals its inverse.

An invertible matrix, or a non-singular matrix, is one that has an inverse. This means that there exists another matrix that, when multiplied with the original, results in the identity matrix. A key condition for a matrix to be invertible is that its determinant must be non-zero. Moreover, the rows (or columns) of the matrix must be linearly independent, indicating that no row (or column) can be written as a linear combination of the others.

In the situation described by the exercise, the matrix P is invertible, which is essential for the diagonalization process to proceed. The invertible nature of P facilitates the final proof where we can manipulate the equations to show that A and its inverse A^-1 are equivalent because the same matrix P is used to diagonalize A.