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Suppose $A$ is $n \times n$ and the equation $A \mathbf{x}=\mathbf{b}$ has a solution for each $\mathbf{b}$ in $\mathbb{R}^{n} .$ Explain why $A$ must be invertible. [Hint: Is A row equivalent to $I_{n} ? ]$

So, that $A$ is a row equivalent to $I _ { n }$ . Hence it is invertible.

Algebra

Chapter 2

Matrix Algebra

Section 2

The Inverse of a Matrix

Introduction to Matrices

Bilal R.

February 13, 2022

If A is 3×3 matrix, then det (2A)= 2 det (A)

Campbell University

University of Michigan - Ann Arbor

Idaho State University

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say we have the system A X equals B and disease Stoma solutions for every vector be in RN Now, if a he's not in veritable, then we reduce the system metrics, eh? Me to something like upper triangle medics here and then on entire roll off zero at the end and appear this whatever. And then here we have the transformer coefficients so easy A is not in veritable. But that means that for some be hence be teal that the system cannot have solution against assumption that the system had solutions for every vector B. And of course, the problem is that we're think we're assuming that B is not in veritable so for the system to have solution always, the medics say, has to be in vertebral.

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