Remind yourself using the row-times-column pattern of matrix multiplication why \(I A = A\) is true.
(b)
Notice how \(E\) is only one entry different from \(I\text{.}\) How does this change the process of computing \(E A\) compared to computing \(I A\text{?}\)
Think of multiplication by \(E\) as “transforming” \(A\) into the result matrix \(E A\text{.}\) How could you describe the transformation in this particular example?
Hint.
In the “transformation” \(A \to E A\text{,}\) which rows of \(A\) stay the same, and which rows change? For the rows that change, how exactly do they change?
(c)
Do you think the same thing will happen when computing \(E\) times some other matrix?
(d)
We know that \(E I = E\text{.}\) But then consider \(E I\) in terms of the first two parts of this discovery activity. So in terms of row operations, what is the relationship between \(E\) and \(I\text{?}\)
Discovery6.2.
Create a \(3 \times 3\) matrix \(E'\) so that for every \(3\times n\) matrix \(A\text{,}\) the result of \(E' A\) is the same as performing the row operation “multiply row \(3\) by \(-4\)” on \(A\text{.}\)
Create a \(3\times 3\) matrix \(E''\) so that for every \(3 \times n\) matrix \(A\text{,}\) the result of \(E'' A\) is the same as performing the row operation “swap rows \(1\) and \(2\)” on \(A\text{.}\)
Matrices \(E,E',E''\) from the discovery activities so far are called elementary matrices. As the preceding activities demonstrate, every elementary row operation has a corresponding elementary matrix.
Discovery6.4.
Suppose we were to take a \(3 \times \ell\) matrix \(A\) and compute
\begin{equation*}
E'' E' E A = E'' \bbrac{ E' (E A) } \text{,}
\end{equation*}
where \(E, E', E''\) are as in Activities 6.1–6.3. How can we interpret this matrix multiplication result in terms of row operations? (Careful of the order of operations!)
Determine elementary matrices \(E_1,E_2,E_3\) so that \(E_3 E_2 E_1 B\) is the identity matrix.
(b)
The matrix \(B\) happens to be invertible. Manipulate the formula \(E_3 E_2 E_1 B = I\) algebraically to obtain a formula for \(\inv{B}\) involving your elementary matrices.
(c)
Tack an identity matrix \(I\) onto the right end of your formula for \(\inv{B}\) from Task b. (Recall that multiplying by \(I\) has no effect.)
Using this new, modified formula for \(\inv{B}\) as inspiration, come up with a procedure to use only row operations (and not elementary matrices) to compute the inverse of a square matrix.
Hint.
Where did your elementary matrices \(E_1,E_2,E_3\) come from? And what are they now “doing” to the identity matrix, and in what order?
Discovery6.6.
Consider the general \(2\times 2\) matrix \(A = \left[\begin{smallmatrix}a \amp b\\c \amp d\end{smallmatrix}\right]\text{.}\)
(a)
Assume that \(a \neq 0\text{.}\) Use the method you developed in Discovery 6.5 to determine the inverse of \(A\text{.}\)
(b)
Where there any other assumptions about the entries of \(A\) (besides \(a \neq 0\)) that you needed to make for this to work? Why?
Hint.
Division by zero is undefined.
(c)
Repeat for the other case: assume \(a=0\text{.}\)
Discovery6.7.
Complete the following tasks for each of the three types of elementary row operations, one at a time:
swap two rows;
multiply a row by a nonzero constant;
add a multiple of one row to another.
(a)
Suppose someone has performed the row operation you are currently considering on a matrix:
\begin{equation*}
A \xrightarrow[\text{op}]{\text{row}} A' \text{.}
\end{equation*}
If you know only the operation and the result \(A'\text{,}\) how can you recover the original matrix \(A\text{?}\)
\begin{equation*}
A' \xrightarrow{\text{?}} A
\end{equation*}
\begin{equation*}
I \xrightarrow[\text{op}]{\text{row}} E I \xrightarrow{\text{(a)}} E' E I \text{,}
\end{equation*}
where
(a) is the same “reverse” row operation you came up with in Task a
\(E\) is the elementary matrix corresponding to the original row operation you are currently considering
and \(E'\) is the elementary matrix corresponding to the (a) row operation.
According to Task a, what should the final result \(E' E I\) be? What does this say in general about the inverse of an elementary matrix of the type you are currently considering?
