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Unformatted text preview: Math 20F Linear Algebra Lecture 8: 2.2 The inverse of a matrix. Definition : The identity matrix is I = [ ij ], where ij = 1 if i = j and ij = 0 if i 6 = j : I = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , in case 4 4 . Definition : The transpose A T is the matrix with rows and columns interchanged, ( A T ) ij =( A ) ji Example: If A = 1 2 3- 2 0- 1 4 5 2 then A T = 1- 2 4 2 5 3- 1 2 . Which is true? ( AB ) T = A T B T or ( AB ) T = B T A T ? An n n matrix A is said to be invertible if there is an n n matrix A- 1 such that (2.2.1) A- 1 A = AA- 1 = I where I is the identity matrix. The matrix A- 1 called the inverse of A is unique. In fact if BA = I then B = BI = B ( AA- 1 ) = ( BA ) A- 1 = IA- 1 = A- 1 . Not all n n matrices are invertible. A matrix which is not invertible is called singular . An invertible matrix is called nonsingular . An inverse of a transformation x T ( x ) is a transformation which takes you back T ( x ) x . The condition A- 1 A = I says that the inverse of the linear transformation x A x is the linear transformation y A- 1 y . In fact, if we compose x A x with...
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