Question

Consider the matrix

$\mathbf{E}\mathbf{=}\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ 0& 0& 1\end{array}\right]$

and an arbitrary$\mathbf{3}\mathbf{\times }\mathbf{3}$matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right]$

Compute EA. Comment on the relationship between A and E A, in terms of the technique of elimination we learned in Section 1.2.

b. Consider the matrix

$\mathbf{E}\mathbf{=}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1/4& 0\\ 0& 0& 1\end{array}\right]$

and an arbitrary $\mathbf{3}\mathbf{\times }\mathbf{3}$matrix A. Compute E A. Comment on the relationship between A and E A.

c. Can you think of a $\mathbf{3}\mathbf{\times }\mathbf{3}$matrix Esuch that E A is obtained from A by swapping the last two rows (for $\mathbf{3}\mathbf{\times }\mathbf{3}$matrix A)?

d. The matrices of the forms introduced in parts (a), (b), and (c) are called elementary: An$\mathit{n}\mathbf{\times }\mathit{n}$matrix Eis elementary if it can be obtained from${\mathit{l}}_{\mathbf{n}}$by performing one of the three elementary row operations on${\mathit{l}}_{\mathbf{n}}$. Describe the format of the three types of elementary matrices.

Short answer

a) The matrix EA is obtained from A by subtract three times the first row of the matrix A from the second row of it ( an elementary row operation).

$EA=\left[\begin{array}{ccc}a& b& c\\ d-3a& e-3b& f-3c\\ g& h& k\end{array}\right]$

b) The matrix EA is obtained from A by dividing the second row of A by 4 (an elementary row operation).

$EA=\left[\begin{array}{ccc}a& b& c\\ \frac{1}{4}d& \frac{1}{4}e& \frac{1}{4}f\\ g& h& k\end{array}\right]$

c) EA is obtained from A by interchanging the last two rows from any 3×3 matrix A.

$EA=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right]$

d) An elementary nn matrix E has the same form as $l_n$except that either

role="math" localid="1659863673074" $$\begin{gathered} 1.{\mathrm{e}}_{\mathrm{ij}}=\mathrm{k}(\ne 0)\mathrm{for}\mathrm{some}\mathrm{i}\ne \mathrm{j}, \\ 2.{\mathrm{e}}_{\mathrm{ij}}=\mathrm{k}(\ne 0,1)\mathrm{for}\mathrm{some}\mathrm{i}, \\ 3.{\mathrm{e}}_{\mathrm{ij}}={\mathrm{e}}_{\mathrm{ij}}=1,{\mathrm{e}}_{\mathrm{ij}}={\mathrm{e}}_{\mathrm{ij}}=0\mathrm{for}\mathrm{some}\mathrm{i}\ne \mathrm{j}, \end{gathered}$$

Step-by-step solution

Given

$\mathrm{E}=\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ 0& 0& 1\end{array}\right]$

and

$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right]$

Find EA

$$\begin{gathered} EA=\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right] \\ EA=\left[\begin{array}{ccc}a& b& c\\ d-3a& e-3b& f-3c\\ g& h& k\end{array}\right] \end{gathered}$$

So the matrix EA is obtained from A by subtract three times the first row of the matrix A from the second row of it (an elementary row operation).

For (b) Find  EA

$$\begin{gathered} E=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1/4& 0\\ 0& 0& 1\end{array}\right] \\ EA=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1/4& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right] \\ EA=\left[\begin{array}{ccc}a& b& c\\ \frac{1}{4}d& \frac{1}{4}e& \frac{1}{4}f\\ g& h& k\end{array}\right] \end{gathered}$$

The matrix EA is obtained from A by dividing the second row of A by 4(an elementary row operation).

For (c) Find  EA

$$\begin{gathered} E=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \\ EA=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right] \\ EA=\left[\begin{array}{ccc}a& b& c\\ g& h& k\\ d& e& f\end{array}\right] \end{gathered}$$

In above case EA is obtained from A by interchanging the last two rows from any 3×3 matrix A.

Answer for (d)

An nn matrix E is elementary if it can be obtained from $l_n$by performing one of the three row operations on $l_n$. The format of three types of elementary matrices are given below.

An elementary nn matrix E has the same form as $l_n$except that either

$$\begin{gathered} 1.{\mathrm{e}}_{\mathrm{ij}}=\mathrm{k}(\ne 0)\mathrm{for}\mathrm{some}\mathrm{i}\ne \mathrm{j}, \\ 2.{\mathrm{e}}_{\mathrm{ij}}=\mathrm{k}(\ne 0,1)\mathrm{for}\mathrm{some}\mathrm{i}, \\ 3.{\mathrm{e}}_{\mathrm{ij}}={\mathrm{e}}_{\mathrm{ij}}=1,{\mathrm{e}}_{\mathrm{ij}}={\mathrm{e}}_{\mathrm{ij}}=0\mathrm{for}\mathrm{some}\mathrm{i}\ne \mathrm{j}, \end{gathered}$$

The final answer

a) The matrix EA is obtained from A by subtract three times the first row of the matrix A from the second row of it ( an elementary row operation).

$EA=\left[\begin{array}{ccc}a& b& c\\ d-3a& e-3b& f-3c\\ g& h& k\end{array}\right]$

b) The matrix EA is obtained from A by dividing the second row of A by 4 (an elementary row operation).

$EA=\left[\begin{array}{ccc}a& b& c\\ \frac{1}{4}d& \frac{1}{4}e& \frac{1}{4}f\\ g& h& k\end{array}\right]$

c) EA is obtained from A by interchanging the last two rows from any 3×3 matrix A.

$EA=\left[\begin{array}{ccc}a& b& c\\ g& h& k\\ d& g& f\end{array}\right]$

d) An elementary nn matrix E has the same form as $l_n$ except that either

role="math" localid="1659863853658" $$\begin{gathered} 1.{\mathrm{e}}_{\mathrm{ij}}=\mathrm{k}(\ne 0)\mathrm{for}\mathrm{some}\mathrm{i}\ne \mathrm{j}, \\ 2.{\mathrm{e}}_{\mathrm{ij}}=\mathrm{k}(\ne 0,1)\mathrm{for}\mathrm{some}\mathrm{i}, \\ 3.{\mathrm{e}}_{\mathrm{ij}}={\mathrm{e}}_{\mathrm{ij}}=1,{\mathrm{e}}_{\mathrm{ij}}={\mathrm{e}}_{\mathrm{ij}}=0\mathrm{for}\mathrm{some}\mathrm{i}\ne \mathrm{j}, \end{gathered}$$