2.5 Matrix Inversion Definition A square matrix A is said to be nonsingular or invertible if there exist a unique matrix(square) B such that AB= BA The matrix b is said to be a multiplicative inverse of A We will refer to the multiplicative inverse of a nonsingular matrix a as simply the inverse of A and denote it by a Some computational results involving inverse are A (AB)-1 when both inverse matrices exist. Finally, if A is symmetric, then a-1is also symmetric Suppose that a, b and a+b are all m x m nonsingular matrices. Then (A+B)1=A-1-A-1(B-1+A-1)- 2.6 A useful idempotent matrix Definition An idempotent matrix is the one that is equal to its square, that is M2=MM M A useful idempotent matrix we will often face is the matrix Mo=I-ii,2.5 Matrix Inversion Definition: A square matrix A is said to be nonsingular or invertible if there exist a unique matrix (square) B such that AB = BA = I. The matrix B is said to be a multiplicative inverse of A. We will refer to the multiplicative inverse of a nonsingular matrix A as simply the inverse of A and denote it by A−1 . Some computational results involving inverse are |A−1 | = 1 |A| , (A−1 ) −1 = A, (A−1 ) ′ = (A′ ) −1 (AB) −1 = B −1A−1 , when both inverse matrices exist. Finally, if A is symmetric, then A−1 is also symmetric. Lemma: Suppose that A, B and A + B are all m × m nonsingular matrices. Then (A + B) −1 = A−1 − A−1 (B −1 + A−1 ) −1A−1 . 2.6 A useful idempotent matrix Definition: An idempotent matrix is the one that is equal to its square, that is M2 = MM = M. A useful idempotent matrix we will often face is the matrix M0 = I − 1 n ii′ , 4