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Unless otherwise specified, assume that all matrices in these exercises are $n \times n .$ Determine which of the matrices in Exercises $1-10$ are invertible. Use as few calculations as possible. Justify your answers.$$\left[\begin{array}{rrr}{5} & {0} & {0} \\ {-3} & {-7} & {0} \\ {8} & {5} & {-1}\end{array}\right]$$

There are 3 pivots in $3 \times 3$ matrix so the matrix is invertible

Algebra

Chapter 2

Matrix Algebra

Section 3

Characterizations of Invertible Matrices

Introduction to Matrices

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

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Unless otherwise specified…

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In Exercises $5-8,$ determ…

in this exercise were considering a three by three matrix A that's provided here, and our job is to determine if they could be in vertebral. One way we could look at this is consider the transpose of a This is a worthwhile approach, since if a is in vertebral, so is a transposed and vice versa. So the transpose of our particular matrix will take Row one and write it as a column 500 Then for Row two, we have column to negative three negative seven and zero finally are less. Column is 85 and negative. One. Notice that the transpose of the Matrix A is now in echelon form, whereas a itself was not. And we see the A transpose has ah pivot in every row. What this means is that a transposed inverse exists. But as we said before, if the inverse of a transpose exists, then that implies immediately that a inverse exists as well. So in this case, were able to determine that a inverse exists without using any row operations whatsoever.

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