1
$\begingroup$

Suppose I need to evaluate the expression $\begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} s+2 & -1 & 0 & 0\\ 0 & s+3 & 0 & 0\\ -1 & -2 & s & -1\\ 2 & -1 & 1 & 4 \end{bmatrix}^{-1} \begin{bmatrix} 0\\ 1\\ 1\\ 1 \end{bmatrix} $ If I compute the inverse of the $4 \times 4$ matrix first, then perform the matrix-vector multiplication, I'm left with the result $ \displaystyle \frac{1}{s+3}$ If instead I perform the matrix-vector multiplication first, keeping in mind that I need to compute the inverse of whatever I'm left with, I get $(s+3)^{-1} = \displaystyle \frac{1}{s+3}$

My question is this: is it safe to assume that I can always perform the matrix-vector multiplication first and then compute the inverse of the resulting scalar (assuming, of course, that the result is invertible) or did I just happen to get lucky in this case?

  • 0
    This will only work if $AB^{-1} = (AB)^{-1} = B^{-1}A^{-1}.$ For example if $A^{-1}$ is similar to $A$.2012-03-12

1 Answers 1

1

The value of $ \begin{bmatrix} s+2 & -1 & 0 & 0\\ 0 & s+3 & 0 & 0\\ -1 & -2 & s & -1\\ 2 & -1 & 1 & 4 \end{bmatrix}^{-1}$ $\text{is}$ $\begin{bmatrix} \frac{1}{s + 2} & \frac{1}{s^2 + 5 s + 6} & 0 & 0 \\ 0 & \frac{1}{s + 3} & 0 & 0\\ \frac{2}{4 s^2 + 9 s + 2} & \frac{9 s + 20}{4 s^3 + 21 s^2 + 29 s + 6} & \frac{4}{4 s + 1} & \frac{1}{4 s + 1}\\ -\frac{2 s + 1}{4 s^2 + 9 s + 2} & -\frac{ - s^2 + 2 s + 5}{4 s^3 + 21 s^2 + 29 s + 6} & -\frac{1}{4 s + 1} & \frac{s}{4 s + 1} \end{bmatrix}$ Multiplying with the row vector retains the 2nd row and eliminates all others.

$\begin{bmatrix} 0 & \frac{1}{s + 3} & 0 & 0\\ \end{bmatrix}$

Multiplying with the column vector picks up the last 3 columns and eliminates others

Answer = $\dfrac{1}{s+3}$

This is, however, merely a coincidence. This is not true in general.

  • 0
    It's actually the state space representation of a linear system. I did find one error in my A-matrix (4x4), though it doesn't change the result that the inverse must be computed first. (The (4,4) entry should read s+4)2012-03-17