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4.19.1. Theorem 4.20 Definition: Nonsingular Matrix Let A aij be a square n n matrix. If there exists another n n matrix B such that BA I , where I is the n n identity matrix, the A is said to be nonsingular and B is called the left-inverse of A. Theorem 4.20. (a) If B is a left inverse of an n n matrix A, then it is also a right inverse, i.e., AB I . Furthermore, B is unique. (b) If B is a right inverse of an n n matrix A, then it is also a left inverse. A is nonsingular. Hence, Proof of (a) Consider the linear transformation T : R n R n such that m T A aij relative to the basis of unit coordinate vectors. Given x such that T x O , let X be the n 1 column matrix that corresponds to x. We have AX 0 , where 0 is the zero column matrix. Thus, B AX 0 for any n n matrix B. If B is a left inverse of A, then B AX BA X I X X 0 This means T x O implies x O . Hence, by theorems 4.8 and 4.10, T is invertible so that there is a unique left inverse T 1 , which is also a right inverse. Matrix representation of the equation TT 1 T 1T I then gives A m T 1 m T 1 A I Multiplying by the left inverse B, we have m T 1 B so that AB I i.e., B is also a right inverse. Finally, let C be another left inverse so that CA I Multiplying this on the right by B gives C B so that the left inverse is unique. QED. Proof of (b) The condition AB I that B is a right inverse of A can be interpreted as saying B has a left inverse A. According to (a), A is also a right inverse of B so that BA I . QED.