1. .
A thread of related interest: http://www.mathhelpforum.com/math-he...-inverses.html
2. Do you know what happens to when you multiply a row of by ? Now note that every row of is multiplied by ....
1. If A is a square matrix satisfying , where I is the identity matrix with the same dimensions as A, does A have an inverse?
I know the answer is yes, because I found the inverse, , basically by trial and error.
But is there a more elegant/systematic way to go about this in general?
2. If A is a square matrix with determinant d, & is a number, what is the determinant of ?
I know the answer, but I don't know why it is..
Thanks for any help
1. .
A thread of related interest: http://www.mathhelpforum.com/math-he...-inverses.html
2. Do you know what happens to when you multiply a row of by ? Now note that every row of is multiplied by ....
Using the Cayley-Hamilton theorem you can use the expression in the first part of you question to find the characteristic polynomial. Now seeing as how it is only a quadratic you should have no problem finding the root. So from here you should be able to know the determinant and if it is in fact invertible.
The determinant of a square matrix can be written as the sum of products of distinct terms at a time multiplied by + or - 1 (which elements appear in the products does not matter for the purposes of answering the question) now as each element is multiplied by the determinant is multiplied by .
CB