Matrix Inverses and Functions in Mathematics
Explore the concept of matrix inverses, inverse functions, invertibility of matrices, and the properties of invertible matrices in mathematics. Learn about the relationships between functions and their inverses, as well as the conditions for a matrix to be invertible or singular. Discover the implications of non-square matrices on invertibility and the properties of matrix products and transposes in relation to invertibility.
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Presentation Transcript
Matrix Inverse Matrix Inverse
Inverse of Function Two functions f and g are inverse of each other f=g-1, g=f-1 For ??? ? ? = ? ? ? g ? = ? ? f ? = ? ? = ? ? f ? g ? = ? ? ? = ?
Inverse of Matrix A and B are inverses to each other ?? = ? For ??? ? ? = ?? ? A B ? = ?? ? = ? B ? = ?? ? A ? = ?? ? = ? ?? = ?
Invertible = Non-singular Not Invertible = Singular Inverse of Matrix A is called invertible if there is a matrix B such that ?? = ? and ?? = ? ? = ? 1 ? = ? 1 B is an inverse of A ? =1 2 5 ? = 5 2 ?? =1 0 1 ?? =1 0 1 3 3 1 0 0 Non-square matrix cannot be invertible
Inverse of Matrix Non-square matrix cannot be invertible? 2d 3d 2d ? ? A B ? = ?? n dim m dim m dim m x n n x m 3d 2d 3d B ? A ? ? = ?? n dim m dim n dim n x m m x n
Inverse of Matrix Not all the square matrix is invertible Unique ?? = ? ?? = ? ?? = ? ?? = ? ? = ?? = ? ?? = ?? ? = ?? = ?
Inverse for matrix product A and B are invertible nxn matrices, is AB invertible? yes ?? 1= ? 1? 1 ? 1? 1?? = ? 1? 1? ? = ? 1? = ? ?? ? 1? 1= ? ?? 1? 1 = ? ? 1 = ? Let ?1,?2, ,??be nxn invertible matrices. The product ?1?2 ??isinvertible, and 1= ?? 1?? 1 1 ?1 1 ?1?2 ??
Inverse for matrix transpose If A is invertible, is AT invertible? ?? 1=? ? 1 ? ???= ???? ? 1? = ? ? 1??= ? ??? 1 ?= ? ?? 1 ?= ? ? 1 ???= ? ?? 1= ?
