-1 is not invertible. 2 2. Use Definition 2.7.1 O to show that -2 Definition 2.7.1....

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-1 is not invertible. 2 2. Use Definition 2.7.1 O to show that -2 Definition 2.7.1. Let A be an n x n matrix. If there exists an n x n matrix B such that AB = In, then A is said to be invertible (or nonsingular) and B is called the inverse of A. We write B= A¯1. If A is not invertible, then A is said to be singular. We only introduce the notion of inverse matrices for square matrices. Therefore, if a matrix is not a square matrix, then it has no inverses. By Definition 2.7.1 O, if A is invertible, then AA- = In. Moreover, A is not invertible if and only if for eachn x n matrix B, AB+ In. We state the following theorem without proof, which provides some equivalent results for inverse matrices. -1 is not invertible. 2 2. Use Definition 2.7.1 O to show that -2 Definition 2.7.1. Let A be an n x n matrix. If there exists an n x n matrix B such that AB = In, then A is said to be invertible (or nonsingular) and B is called the inverse of A. We write B= A¯1. If A is not invertible, then A is said to be singular. We only introduce the notion of inverse matrices for square matrices. Therefore, if a matrix is not a square matrix, then it has no inverses. By Definition 2.7.1 O, if A is invertible, then AA- = In. Moreover, A is not invertible if and only if for eachn x n matrix B, AB+ In. We state the following theorem without proof, which provides some equivalent results for inverse matrices.