My Differential Equations book is going over finding the inverse of Matrices, and clearly I've forgotten my college algebra. I have no idea why this works.
The first example gives this:
$\begin{bmatrix}1 & -1 & -1\\3 & -1 & 2\\2 & 2 & 3\end{bmatrix}$ to give the identity matrix, which is of course $\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}$
The book says: Obtain zeros in the off diagonal position in the first (-3) times the first row to the second row and adding -2 times the first row to the third row. This generates the Matrices:
$\begin{bmatrix}1 & -1 & -1\\0 & 2 & 5\\0 & 4 & 5\end{bmatrix} \begin{bmatrix}1 & 0 & 0\\-3 & 1 & 0\\-2 & 0 & 1\end{bmatrix}$
It Continues: Obtain a one in the diagonal position of the second column by multiplying the second row by $\frac{1}{2}$.
$\begin{bmatrix}1 & -1 & -1\\0 & 1 & \frac{5}{2}\\0 & 4 & 5\end{bmatrix} \begin{bmatrix}1 & 0 & 0\\-\frac{3}{2} & \frac{1}{2} & 0\\-2 & 0 & 1\end{bmatrix}$
And it proceeds: Obtain zeros in the off-diagonal positions inthe second column by adding the second row to the first row and adding (-4) times the second row to the third row.
$\begin{bmatrix}1 & 0 & \frac{3}{2}\\0 & 1 & \frac{5}{2}\\0 & 0 & -5\end{bmatrix} \begin{bmatrix}-\frac{1}{2} & \frac{1}{2} & 0\\-\frac{3}{2} & \frac{1}{2} & 0\\4 & -2 & 1\end{bmatrix}$.
This is where the set of rules I was constructing to repeat this breaks down completely. However, in the first step, why are we allowed to add a number to the first column of a row being operated on in the left matrix to the second matrix, then multiply it by an arbitrary row? Then How is simply multiplying across rows simultaneously by a constant multiplicatively consistent? and worst of all in the final operation shown, why is the 4 not negative? Why is anything other than the first column of the row being operated on being modified, unlike the other two additive operations that used a coefficient?
I tried to show this for myself by constructing a matrix to represent $\bf{AB}$. One matrix was constructed of elements $a_1,b_1...$ proceeding first down rows, not columns, and the second of such elements $a_2,b_2...$
It looked like thus:
\begin{bmatrix}a_1a_2+b_2d_2+c_2g_2 & a_1b_2+b_1e_2 + c_1h_1 & a_1c_2+b_1f_2+c_1i_2\\... & ... & ...\\... & ... & ...\end{bmatrix}
I would like to learn how to do this in row operations, because, well, it's faster, and several of these may be on the test. However, in essence, all of the operations in the procedure seem arbitrary and I can't construct rules for repeating it. Thank you for any help.
By the way, the book is:Elementary Differential Equations, Ninth Edition, By Boyce and DiPrima. The book is currently in it's tenth edition.
