# Thread: Matrix inverses, determinants... questions

1. ## Matrix inverses, determinants... questions

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

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

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

Thanks for any help

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

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

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

Thanks for any help
1. $A^2 +10A = I \Rightarrow A^{-1} A^2 +10 A^{-1} A = A^{-1} I \Rightarrow A + 10 I = A^{-1}$.

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

2. Do you know what happens to $\det(A)$ when you multiply a row of $A$ by $\lambda$? Now note that every row of $A$ is multiplied by $\lambda$ ....

3. 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.

4. Originally Posted by Aileys.
1. If A is a square matrix satisfying $A^2 +10A = I$, 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, $10I + A$, basically by trial and error.
You don't need trial and error: $A^2+ 10A= A(A+ 10I)= (A+ 10I)A= I$ immediately tells you that A+ 10I is the inverse of A.

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

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.

5. Originally Posted by Aileys.

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

I know the answer, but I don't know why it is..
The determinant of a $n\times n$ square matrix can be written as the sum of products of $n$ 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 $\lambda$ the determinant is multiplied by $\lambda^n$.

CB