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2.5 Inverse Matrices - MIT Mathematics

Inverse Inverse Matrices '&$%1If the square matrixAhas an Inverse , then bothA 1A=IandAA 1= test invertibility is elimination :Amust haven(nonzero) for invertibility is the determinant ofA:detAmust not be tests for invertibility isAx=0:x=0must be the only (same size) are invertible then so isAB:|(AB) 1=B 1A 1=Iisnequations forncolumns ofA 1. Gauss-Jordan eliminates[A I]to[I A 1].7 The last page of the book gives14equivalent conditions for a squareAto be a square matrix. We look for an Inverse matrix A 1of the same size,such thatA 1timesAequalsI.

2.5. Inverse Matrices 83 2.5 Inverse Matrices 1 If the square matrix A has an inverse, then both A−1A = I and AA−1 = I. 2 The algorithm to test invertibility is elimination: A must have n (nonzero) pivots. 3 The algebra test for invertibility is the determinant of A: detA must not be zero.

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