Math Problem Statement

Write the elementary row operation corresponding to the elementary matrix and find its inverse.

Solution

Let's break down the problem presented in the image.

Part (b):

You are asked to write the elementary row operation corresponding to the elementary matrix EE, which is given by:

1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 12 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ This matrix is almost the identity matrix, except for the entry $$12$$ in the $$(3,2)$$-position. This indicates that the elementary row operation associated with this matrix is: $$R_3 \rightarrow R_3 + 12R_2$$ This operation means that we add 12 times the second row to the third row. ### Part (c): You are asked to find the inverse of the matrix $$E$$. To find the inverse of an elementary matrix, you reverse the elementary row operation it represents. Since the matrix $$E$$ represents the operation $$R_3 \rightarrow R_3 + 12R_2$$, the inverse of this matrix will represent the reverse of this operation, which is: $$R_3 \rightarrow R_3 - 12R_2$$ Thus, the inverse matrix $$E^{-1}$$ is: $$E^{-1} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -12 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ This completes the solution. Would you like further clarification on any step? Here are 5 related questions to expand your understanding: 1. What is the general method for finding the inverse of an elementary matrix? 2. How do different types of elementary row operations (swapping, scaling, adding rows) affect the inverse matrix? 3. What is the significance of elementary matrices in solving systems of linear equations? 4. How does performing a sequence of row operations relate to matrix multiplication? 5. How can you check if a matrix is invertible by looking at its elementary row operations? **Tip:** Remember that elementary matrices are closely linked to row operations. Each elementary matrix corresponds to a specific operation, and its inverse represents the reverse of that operation.

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Math Problem Analysis

Mathematical Concepts

Elementary Matrices
Row Operations
Matrix Inversion

Formulas

Elementary matrix operation: R3 → R3 + kR2
Matrix inversion: Reverse of elementary operations

Theorems

Elementary Matrix Theorem
Inverse Matrix Theorem

Suitable Grade Level

Undergraduate (Linear Algebra)

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