Invertable Matrix Math

The inverse of a square matrix a sometimes called a reciprocal matrix is a matrix a 1 such that aa 1 i 1 where i is the identity matrix. A square matrix a has an inverse iff the determinant a 0 lipschutz 1991 p. Courant and hilbert 1989 p.

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10 use the notation a to denote the inverse matrix.

Invertable matrix math. This section consists of a single important theorem containing many equivalent conditions for a matrix to be invertible. A 1 a i. Dave4math mathematics invertible matrix and it s properties. A a 1 i.

We will append two more criteria in section 5 1. Furthermore the following properties hold for an invertible matrix a. Same thing when the inverse comes first. A 1 1 a.

1 8 8 1. Ka 1 k 1a 1 for nonzero scalar k. For any invertible n by n matrices a and b. The invertible matrix theorem.

Ax x a 1 if a has orthonormal columns where denotes the moore penrose inverse and x is a vector. This is one of the most important theorems in this textbook. Okay so you know what a linear transformation is but what exactly is an invertible linear transformation. When we multiply a number by its reciprocal we get 1.

8 1 8 1. At 1 a 1 t. When we multiply a matrix by its inverse we get the identity matrix which is like 1 for matrices. In this article i cover invertible matrices and work through several examples.