Find the inverse of the elementary matrix.
 
 
1

0

*k*

1
 
 
,   *k* ≠ 0

To find the inverse of the elementary matrix:

\[
E = \begin{pmatrix} 
1 & 0 \\
k & 1 
\end{pmatrix}
\]

where \( k \neq 0 \), we need to find a matrix \( E^{-1} \) such that:

\[
E E^{-1} = I
\]

where \( I \) is the identity matrix:

\[
I = \begin{pmatrix} 
1 & 0 \\
0 & 1 
\end{pmatrix}
\]

Assume the inverse matrix \( E^{-1} \) has the form:

\[
E^{-1} = \begin{pmatrix} 
a & b \\
c & d 
\end{pmatrix}
\]

Now, multiply \( E \) and \( E^{-1} \):

\[
E E^{-1} = \begin{pmatrix} 
1 & 0 \\
k & 1 
\end{pmatrix} \times \begin{pmatrix} 
a & b \\
c & d 
\end{pmatrix} = \begin{pmatrix} 
1a + 0c & 1b + 0d \\
ka + 1c & kb + 1d 
\end{pmatrix}
\]

Simplifying, we get:

\[
E E^{-1} = \begin{pmatrix} 
a & b \\
ka + c & kb + d 
\end{pmatrix}
\]

For this product to equal the identity matrix \( I \), we need:

\[
\begin{pmatrix} 
a & b \\
ka + c & kb + d 
\end{pmatrix} = \begin{pmatrix} 
1 & 0 \\
0 & 1 
\end{pmatrix}
\]

This gives us the following system of equations:

1. \( a = 1 \)
2. \( b = 0 \)
3. \( ka + c = 0 \)
4. \( kb + d = 1 \)

From these equations:

- From equation 1, \( a = 1 \).
- From equation 2, \( b = 0 \).
- Substitute \( a = 1 \) into equation 3: \( k(1) + c = 0 \) implies \( c = -k \).
- Substitute \( b = 0 \) into equation 4: \( k(0) + d = 1 \) implies \( d = 1 \).

So, the inverse matrix \( E^{-1} \) is:

\[
E^{-1} = \begin{pmatrix} 
1 & 0 \\
-k & 1 
\end{pmatrix}
\]

### Conclusion

The inverse of the elementary matrix \( E = \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} \) is:

\[
E^{-1} = \begin{pmatrix} 
1 & 0 \\
-k & 1 
\end{pmatrix}
\]

Would you like more details or have any questions?

Here are 8 related questions to consider:

1. How do you find the inverse of a 3x3 elementary matrix?
2. What are the properties of an elementary matrix?
3. How does the inverse of a matrix relate to solving systems of linear equations?
4. Why must \( k \neq 0 \) for the inverse to exist?
5. What is the determinant of an elementary matrix?
6. Can every elementary matrix be inverted?
7. What is the geometric interpretation of an elementary matrix?
8. How do you verify that two matrices are inverses of each other?

**Tip:** The inverse of a lower triangular elementary matrix can be easily found by changing the sign of the non-diagonal element.

Learn how to find the inverse of an elementary matrix using detailed step-by-step instructions. This guide covers matrix algebra concepts, including elementary matrices and matrix inverses. Suitable for advanced high school students, college students, and anyone interested in linear algebra.

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