Example 4.1 (Testing for Matrix Invertibility)Let us begin with exploring if a square matrix A is invertible (see Sec-tion 2.2.2). For the smallest cases, we already know when a matrixis invertible. If A is a 1 × 1 matrix, i.e., it is a scalar number, thenA = a =⇒ A−1 = 1a. Thus a 1a = 1 holds, if and only if a 6= 0.For 2 ×2 matrices, by the definition of the inverse (Definition 2.3), weknow that AA−1 = I. Then, with (2.24), the inverse of A is