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b) Show that if A is idempotent, then 2A - I is invertible and is its own inverse. Since there are only 2 idempotent square matrices, you can just try them both for parts a and b. The alternative is to substitute I-A into the definition for idempotent for part a, and calculate (2A-I)(2A-I) using the definition...

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We are to prove that if A, B, and C are invertible matrices, then ABC is invertible and its inverse is . We know that ABC is an matrix because A, B, and C are all matrices. We know that the product of two invertible matrices of the same dimensions is also invertible.

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Lots of answers here, but I think there are still some more things worth saying. It has been noted that [math]AB=AC[/math] is equivalent to [math]A(B-C)=[0][/math].

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Invertible matrix and its inverse. A square matrix A is called invertible or non-singular if there exists a matrix B such that AB = BA = I n, where I n is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A, denoted A −1. Definite matrix

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A product of invertible n x n matrices is invertible, and the inverse of the product is the product of their inverses in the same order. If A can be row reduced to the identity matrix, then A must be invertible.

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where i is the identity matrix. The inverse of a 2×2 matrix take for example an arbitrary 2×2 matrix a whose determinant (ad − bc) is not equal to zero. where a, b, c and d are numbers. The inverse is: the inverse of a general n × n matrix a can be found by using the following equation. where the adj (a) denotes the adjoint of a matrix. It ...

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More generally, if W is any invertible matrix, and λ is an eigenvalue of A with generalized eigenvector v, then (W −1 AW − λI) k W −k v = 0. Thus λ is an eigenvalue of W −1 AW with generalized eigenvector W − k v .