Is Matrix A Invertible and What Are Its Bounds?

[twoski]

Homework Statement

This isn't exactly homework per se, i just am having a hard time figuring out how to solve these types of questions generally...

Consider the 3x3 matrix A = diag[1,3,1]. Show that this matrix is invertible no matter what its dimensions are. Determine an upper bound on ||A^{-1}||_{∞} and on the condition number of A

The Attempt at a Solution

So from what I've picked up on so far, i start out:

A = D + E

where D contains all values of A that are in the same row and column (ie. 1,1... 2,2... etc) and E contains all values of A that are not on the diagonal.

Next, say that B = D^{-1} * E and A = D(I + B)

It follows that ||B||_{∞} < 1 because of diagonal dominance. Hence A is invertible (I have no idea what diagonal dominance means in this context, i just wrote down what I'm supposed to say in order to get marks here - not much explanation has been given on this part).

How do i find the condition number and upper bounds?

 

[twoski]

So apparently the condition number is determined by adding the infinity norm of A with the infinity norm of its inverse - is this correct?