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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.$$\left[\begin{array}{rrr}{0} & {3} & {-5} \\ {1} & {0} & {2} \\ {-4} & {-9} & {7}\end{array}\right]$$

the matrix is not invertible.

Algebra

Chapter 2

Matrix Algebra

Section 3

Characterizations of Invertible Matrices

Introduction to Matrices

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University of Michigan - Ann Arbor

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In mathematics, the absolu…

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Unless otherwise specified…

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In Exercises $5-8,$ determ…

in this example, we have a three by three matrix a provided, and our goal here is to determine if it has an inverse for this particular case. I'm going to take the approach of role operations until it's reduced into echelon form. So to that end, let me make row to our first row, which is 102 row three. Let's make that the second row negative four negative 97 and the old first row could be our new third row 03 and negative five. Now, since we're only going for echelon form from this pivot here will eliminate this entry below, so this will be raw equivalent to first copy 102 then multiply row one by four. At the result to Row two will obtain zero negative nine and let's see eight plus seven, which is a 15 then the next row is 03 and negative five. Notice, however, that the last two rows air now suspicious. If we take negative three times the third row, then we would get to negative nine and positive 15. This means these rows are linearly dependent, and so if we make a new row operation it would be 102 copy zero negative, 9 15 and say Divide that row, Bite negative three or excuse me, divided by three and added to Row three that produce Onley zeros. So that tells us we have a pivot here and here. But the third column is not a pivot column due to this zero. So this tells us the A is not. Let's start with a verb. A is not row equivalent to the identity matrix I. And this implies that a inverse does not exist. If we did get a pivot here, then we would have concluded the opposite. That a inverse does exist because then it would be row equivalent to the identity matrix.

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