Math Problem Statement

cari jawabanya

Ubahlah matriks berikut menjadi matriks identitas dengan OBE

Solution

The uploaded image presents a mathematical exercise asking to transform various matrices into identity matrices using elementary row operations (EROs). Here, I'll provide the row operations needed to convert each matrix to its respective identity matrix.

Matrix (1):

1 & 2 \\ 2 & 5 \end{pmatrix}$$ #### Operations: 1. $$R_2 = R_2 - 2R_1$$ 2. $$R_2 = \frac{1}{R_2}$$ 3. $$R_1 = R_1 - 2R_2$$ #### Result: $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ ### Matrix (2): $$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ #### Operations: 1. $$R_2 = R_2 - 3R_1$$ 2. $$R_2 = \frac{1}{R_2}$$ 3. $$R_1 = R_1 - 2R_2$$ #### Result: $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ ### Matrix (3): $$\begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{pmatrix}$$ #### Operations: 1. $$R_2 = R_2 - R_1$$ 2. $$R_3 = R_3 - R_1$$ 3. $$R_3 = R_3 - 2R_2$$ (after making $$R_2$$ a lead 1) 4. Back substitution to clear above entries. #### Result: $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ ### Matrix (4): $$\begin{pmatrix} 6 & 2 & 5 \\ 1 & 7 & 0 \end{pmatrix}$$ #### Operations: 1. Swap $$R_1$$ and $$R_2$$. 2. $$R_1 = R_1 / R_{1,1}$$ to make the leading coefficient 1. 3. $$R_2 = R_2 - 6R_1$$ to clear the first column. 4. Scale $$R_2$$ to make the pivot 1. 5. Clear above entries in column 2. #### Result: $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ **Do you need more detailed steps for each operation or any clarification on the specific steps?** **Additional Questions:** 1. Do you need a walkthrough on how to perform elementary row operations? 2. Would you like to learn about the conditions for a matrix to be invertible? 3. Are you interested in understanding the geometric interpretation of these transformations? 4. Can I assist you with any other types of matrix operations like multiplication or transposition? 5. Would a summary of matrix row reduction strategies be helpful? **Tip:** Always double-check each step during row operations as small miscalculations can lead to errors in the final result.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrices
Elementary Row Operations

Formulas

Elementary row operations (ERO): row swapping, row multiplication, row addition/subtraction

Theorems

Identity Matrix
Row Reduction to Row Echelon Form

Suitable Grade Level

Undergraduate Level