MA 35100 Lecture Notes - Lecture 25: California State Route 1, Computer-Aided Technologies, If And Only If

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lmrice

23 Nov 2022

School

Purdue University

Department

Mathematics

Course

MA 35100

Professor

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Document Summary

An nxn matrix a is invertible iff there is an uxh matrix (such that. Ac = ca =1( we call can inverse for al. Theorem 3. 9: a is nonsingular iff a is invertible. If a is uxn has a left inverse, i. e. 3 with ca=i, or a right inverse, i. e. d with ad =1, then we can show a is non-singular. Cax= cb, x =cb: x = cb is the unique. A (db):ad (b):b: c. x= db is a solution to. If a is invertible, ac= ca:i (and c =a " as a is nonsingular - Turns out that a" is the only (with cac=ca:i). If you have found either a left inverse car right inverse pfor a, you have found a " 3 satisfies a -sa+131:0 r now a is invertible. Useful, but only when you have a lot of into about a. I3 we can manipulate to solve for the us through rr, referencing matrix multiplication.

Related Questions

11 3 3 (10) 7. Let A = 1-3 -5 -3). Find a nonsingular (invertible) matrix P and a diagonal 3 3 1 / matrix D such that P-AP = D.

3. (7 points) Decide whether the following statements are trueor false (give a short justication if you
want full points!):
(a) A homogeneous system of three equations and four variables hasa nontrivial solution.

(b) If u1; u2 are both solutions to the system Ax = b then(u1+u2)/2 is also a solution.

(d) If A;B;C are nxn non-singular matrices, then (ABC)^-1 =C^-1A^-1B^-1

(e) There can be a 4x4 non-singular matrix A such that Ax = 0 has anon-trivial solution (i.e.
x is a solution with some non-zero entries).

(f) Every square matrix with 1's down the diagonal isinvertible

(g) If A is invertible, then A^2 is invertible too.