Finding the Inverse of a 4x4 Matrix Using Elementary Row Operations

Math Problem Statement

Dapatkan invers dari matriks-matriks berikut ini jika matriksnya adalah matriks invertible dengan menggunakan Operasi Baris Elementer (OBE) dan cek jawabanya dengan mengalikannya dengan matriks awalnya:

[[0, 0, 2, 0], [1, 0, 0, 1], [0, - 1, 3, 0], [2, 1, 5, - 3]]

Solution

Untuk menemukan invers dari matriks $A = \begin{pmatrix} 0 & 0 & 2 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & -1 & 3 & 0 \\ 2 & 1 & 5 & -3 \end{pmatrix}$ , kita akan menggunakan metode Operasi Baris Elementer (OBE).

Langkah-langkahnya adalah sebagai berikut:

Kita gabungkan matriks $A$ dengan matriks identitas 4x4, menghasilkan $[A | I]$ .
Lalu kita lakukan operasi baris elementer untuk mengubah matriks $A$ menjadi matriks identitas.
Setelah itu, matriks di sebelah kanan (yang awalnya adalah matriks identitas) akan menjadi invers dari $A$ .

Mari saya selesaikan untuk mendapatkan invers matriks ini.

Invers dari matriks $A$ adalah:

-0.8 & 0.6 & 0.2 & 0.2 \\ 1.5 & 0 & -1 & 0 \\ 0.5 & 0 & 0 & 0 \\ 0.8 & 0.4 & -0.2 & -0.2 \end{pmatrix}$$ Selanjutnya, mari kita cek hasilnya dengan mengalikan matriks ini dengan matriks awal untuk memastikan bahwa hasilnya adalah matriks identitas. It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can verify the solution by multiplying the inverse matrix $$A^{-1}$$ by the original matrix $$A$$ to check if you get the identity matrix $$I$$. Would you like further assistance on how to do that manually or have more questions about matrix operations? Here are five related questions that might interest you: 1. What are the conditions for a matrix to have an inverse? 2. How do you find the determinant of a matrix and what does it tell you? 3. Can every square matrix be inverted? 4. How can the rank of a matrix affect its invertibility? 5. What are the different methods for finding the inverse of a matrix? **Tip:** The determinant of a matrix can help you determine if the matrix is invertible. If the determinant is zero, the matrix has no inverse.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Elementary Row Operations
Inverse Matrix Verification

Formulas

Inverse matrix formula using row operations
Identity matrix

Theorems

A matrix is invertible if and only if its determinant is non-zero
Elementary row operations can be used to find the inverse of a matrix

Suitable Grade Level

Grades 11-12 / University level

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