Question

Reread the discussion leading to the result given in (7). Does the matrix sI-Aalways have an inverse? Discuss.

Short answer

The matrix sI-A will have an inverse when s has the largest eigenvalue of the matrix A.

Step-by-step solution

Define matrix exponential.

Consider a square matrix A of size n*n. This matrix can contain either complex numbers or real numbers. The matrix can be calculated as

where I is the unit matrix of order n.

Therefore the infinite matrix power series is

Now, the matrix exponential is defined as the sum of the infinite matrix power series. It is denoted by the expression e^At. It is given by the formula,

Find whether the matrix sI-A always have an inverse.

The inverse matrix of a square matrix is its multiplicative inverse. Consider a matrix A. If this matrix has an inverse it is denoted as invertible or non singular.

To find the inverse of matrix A, first you have to find A^-1. When you multiply A and A^-1 the result will be an identity matrix. It is explained by the property,

AA^-1= A^-1 A =I

If,

det(sI-A) = 0

then s is an eigenvalue of A. When s is not an eigenvalue of A then sI-A is invertible.

So take s to be the largest eigenvalue of A then sI-A is invertible.

Hence for sI-A to have an inverse, s has to be the largest eigenvalue of A.