Question

Suppose that A is an n x n matrix such that A² - 2A + Iₙ = 0. Show that A is invertible.

a) Prove A is invertible
b) Use the inverse matrix
c) Substitute values for A
d) Not enough information

Answer 1

To prove that the matrix A is invertible, we rearranged the given equation to suggest that B = -A + 2I_n is the inverse of A because their product yields the identity matrix. Consequently, A is indeed invertible, and this is confirmed without needing to substitute specific values for A.

Step-by-step explanation:

To show that a matrix A is invertible, we need to demonstrate that there exists a matrix B such that AB = BA = In, where In is the identity matrix. Given that A is an n x n matrix satisfying the equation A2 - 2A + In = 0, we can rearrange this equation to A(A - 2In) = -In.

If we multiply both sides by -1, we get A(-A + 2In) = In, which suggests that -A + 2In could be the inverse of A, since their multiplication results in the identity matrix. Thus, we can infer that B = -A + 2In is the inverse of A, and by definition, A is invertible.

To use the inverse matrix B, we would typically perform matrix operations necessitating the inverse, such as solving systems of linear equations.

It is not required to substitute specific values for A to show that it is invertible; the algebraic manipulation based on the given equation is sufficient.

The correct option is a) Prove A is invertible