DLA Discovery guide

Discovery guide 6.1 Discovery guide

Discovery 6.1.

Consider the matrices

$\begin{align*} I \amp = \begin{bmatrix} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \end{bmatrix}, \amp E \amp = \begin{bmatrix} 1 \amp 0 \amp 0 \\ 2 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \end{bmatrix}, \amp A \amp = \left[\begin{array}{rrrr} 1 \amp 0 \amp 2 \amp -1 \\ 1 \amp 2 \amp 3 \amp 4 \\ 0 \amp -1 \amp 0 \amp 3 \end{array}\right]. \end{align*}$

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(a)

Remind yourself using the row-times-column pattern of matrix multiplication why $I A = A$ is true.

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(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?

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(c)

Do you think the same thing will happen when computing $E$ times some other matrix?

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(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 $I\text{?}$

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Discovery 6.2.

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 $A\text{.}$

Hint

What was the pattern you identified in Discovery 6.1.d?

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Discovery 6.3.

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 $A\text{.}$

Hint

What was the pattern you identified in Discovery 6.1.d?

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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.

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Discovery 6.4.

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!)

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Discovery 6.5.

Consider $B = \left[\begin{array}{rrr} 1 \amp 0 \amp -3 \\ 0 \amp 0 \amp 2 \\ 0 \amp 1 \amp 0 \end{array}\right] \text{.}$

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(a)

Determine elementary matrices $E_1,E_2,E_3$ so that $E_3 E_2 E_1 B$ is the identity matrix.

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(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.

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(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?

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Discovery 6.6.

Consider the general $2\times 2$ matrix $A = \left[\begin{smallmatrix}a \amp b\\c \amp d\end{smallmatrix}\right]\text{.}$

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(a)

Assume that $a \neq 0\text{.}$ Use the method you developed in Discovery 6.5 to determine the inverse of $A\text{.}$

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(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.

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(c)

Repeat for the other case: assume $a=0\text{.}$

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Discovery 6.7.

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.

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(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 $A\text{?}$

$\begin{equation*} A' \xrightarrow{\text{?}} A \end{equation*}$

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(b)

Suppose we consider Task a with $A = I\text{:}$

$\begin{equation*} I \xrightarrow[\text{op}]{\text{row}} E I \xrightarrow{\text{(a)}} E' E I \text{,} \end{equation*}$

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?