Math Problem Statement
Find the inverse of the elementary matrix. 1
0
k
1 , k ≠ 0
Solution
To find the inverse of the elementary matrix:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Elementary Matrices
Matrix Inverses
Formulas
-
Theorems
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Suitable Grade Level
Advanced High School and above