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# l8 - Math 20F Linear Algebra Lecture 8 2.2 The inverse of a...

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Math 20F Linear Algebra Lecture 8: 2.2 The inverse of a matrix. Definition : The identity matrix is I = [ δ ij ], where δ ij = 1 if i = j and δ 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 0 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 y A - 1 y we get x A x A - 1 ( A x )=( AA - 1 ) x = I x = x .

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l8 - Math 20F Linear Algebra Lecture 8 2.2 The inverse of a...

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