# Matrix inverses, determinants... questions

• Aug 10th 2009, 05:22 PM
Aileys.
Matrix inverses, determinants... questions
1. If A is a square matrix satisfying $\displaystyle A^2 +10A = I$, 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, $\displaystyle 10I + A$, basically by trial and error.

2. If A is a square matrix with determinant d, & $\displaystyle \lambda$ is a number, what is the determinant of $\displaystyle \lambda A$?

I know the answer, but I don't know why it is..

Thanks for any help
• Aug 10th 2009, 07:10 PM
mr fantastic
Quote:

Originally Posted by Aileys.
1. If A is a square matrix satisfying $\displaystyle A^2 +10A = I$, 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, $\displaystyle 10I + A$, basically by trial and error.

2. If A is a square matrix with determinant d, & $\displaystyle \lambda$ is a number, what is the determinant of $\displaystyle \lambda A$?

I know the answer, but I don't know why it is..

Thanks for any help

1. $\displaystyle A^2 +10A = I \Rightarrow A^{-1} A^2 +10 A^{-1} A = A^{-1} I \Rightarrow A + 10 I = A^{-1}$.

A thread of related interest: http://www.mathhelpforum.com/math-he...-inverses.html

2. Do you know what happens to $\displaystyle \det(A)$ when you multiply a row of $\displaystyle A$ by $\displaystyle \lambda$? Now note that every row of $\displaystyle A$ is multiplied by $\displaystyle \lambda$ ....
• Aug 10th 2009, 07:25 PM
putnam120
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.
• Aug 10th 2009, 09:32 PM
HallsofIvy
Quote:

Originally Posted by Aileys.
1. If A is a square matrix satisfying $\displaystyle A^2 +10A = I$, 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, $\displaystyle 10I + A$, basically by trial and error.

You don't need trial and error: $\displaystyle A^2+ 10A= A(A+ 10I)= (A+ 10I)A= I$ immediately tells you that A+ 10I is the inverse of A.

Quote:

Originally Posted by Aileys.
2. If A is a square matrix with determinant d, & $\displaystyle \lambda$ is a number, what is the determinant of $\displaystyle \lambda A$?

I know the answer, but I don't know why it is..

Thanks for any help

Think about the special case in which A is a diagonal matrix.
• Aug 10th 2009, 11:02 PM
CaptainBlack
Quote:

Originally Posted by Aileys.

2. If A is a square matrix with determinant d, & $\displaystyle \lambda$ is a number, what is the determinant of $\displaystyle \lambda A$?

I know the answer, but I don't know why it is..

The determinant of a $\displaystyle n\times n$ square matrix can be written as the sum of products of $\displaystyle n$ 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 $\displaystyle \lambda$ the determinant is multiplied by $\displaystyle \lambda^n$.

CB