CHAPTER 8: MATRICES and DETERMINANTS

2 5 Inverse Matrices 81 2 5 Inverse Matrices Suppose A is a square matrix We look for an “inverse matrix” A 1 of the same size, such that A 1 times A equals I Whatever A does, A 1 undoes Their product is the identity matrix—which does nothing to a vector, so A 1Ax D x But A 1 might not exist What a matrix mostly does is to multiply

Matrices, transposes, and inverses

Feb 01, 2012 · The notion of an inverse matrix only applies to square matrices - For rectangular matrices of full rank, there are one-sided inverses - For matrices in general, there are pseudoinverses, which are a generalization to matrix inverses Example Find the inverse of A = 11 11 Wehave 11 11 ab cd = 10 01 =⇒ a+cb

Determinants & Inverse Matrices

A matrix has an inverse exactly when its determinant is not equal to 0 ***** *** 2⇥2inverses Suppose that the determinant of the 2⇥2matrix ab cd does not equal 0 Then the matrix has an inverse, and it can be found using the formula ab cd 1 = 1 det ab cd d b ca Notice that in the above formula we are allowed to divide by the determi-

Lecture 6 Inverse of Matrix

De &nition 7 1 A square matrix An£n is said to be invertible if there exists a unique matrix Cn£n of the same size such that AC =CA =In: The matrix C is called the inverse of A; and is denoted by C =A¡1 Suppose now An£n is invertible and C =A¡1 is its inverse matrix Then the matrix equation A~x =~b can be easily solved as follows

LEFT/RIGHT INVERTIBLE MATRICES

Feb 06, 2014 · LEFT/RIGHT INVERTIBLE MATRICES MINSEON SHIN (Last edited February 6, 2014 at 6:27pm ) De nition 1 Let A be an m n matrix We say that A is left invertible if there exists an n m matrix C such that CA = I n (We call C a left inverse of A 1) We say that A is right invertible if there exists an n m matrix D such that AD = I m

matrix identities - New York University

0 10 matrix inversion lemma (sherman-morrison-woodbury) using the above results for block matrices we can make some substitutions and get the following important results: (A+ XBXT) 1 = A 1 A 1X(B 1 + XTA 1X) 1XTA 1 (10) jA+ XBXTj= jBjjAjjB 1 + XTA 1Xj (11) where A and B are square and invertible matrices but need not be of the

56 Using the inverse matrix to solve equations

A is called the matrix of coeﬃcients 2 Solving the simultaneous equations Given AX = B we can multiply both sides by the inverse of A, provided this exists, to give A−1AX = A−1B But A−1A = I, the identity matrix Furthermore, IX = X, because multiplying any matrix by an identity matrix of the appropriate size leaves the matrix

Matrices and Linear Algebra

In this case B is called the inverse of A, and the notation for the inverse is A−1 Examples (i) Let A =} 13 −12] Then A−1 = 1 5} 2 −3 11] (ii) For n =3wehave A = 12−1 −13−1 −23−1 A−1 = 01−1 −13−2 −37−5 A square matrix need not have an inverse, as will be discussed in the next section As examples, the two

Matrix Operations on the TI-89 - homepagesmathuicedu

(rref): Press [2nd][MATH] select [4:Matrix] Select the desired form followed by the name of the matrix and press enter For example: Inverse Matrices: Select the name of the matrix and raise it to the –1 power The matrix A above is not invertible so we consider If you want your results in fractions select [Exact/Approx] after pressing [MODE]