An n × n matrix A is said to be invertible (or nonsingular) if there is another n × n matrix B with the property that
AB = BA = In
where I n denotes the n × n identity matrix. (See Exercise 20.) The matrix B is called an inverse to the matrix A. Exercises 30–38 concern various aspects of matrices and their inverses.
Show that if an n × n matrix A is invertible, then A can have only one inverse matrix. Thus, we may write A−1 to denote the unique inverse of a nonsingular matrix A. (Hint: Suppose A were to have two inverses B and C. Consider B(AC).)
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