Find the Inverse of Matrices Using Gauss-Jordan Method

Math Problem Statement

𝑭𝒊𝒏𝒅 𝑰𝒏𝒗𝒆𝒓𝒔𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒐𝒍𝒍𝒐𝒘𝒊𝒏𝒈 𝑴𝒂𝒕𝒓𝒊𝒄𝒆𝒔 𝒖𝒔𝒊𝒏𝒈 𝑮𝒂𝒖𝒔𝒔 − 𝑱𝒐𝒓𝒅𝒂𝒏 𝑴𝒆𝒕𝒉𝒐𝒅. 𝑩 = [ 𝟎 𝟏 𝟐 𝟐 𝟏 𝟏 𝟐 𝟑 𝟐 𝟐 𝟐 𝟑 𝟐 𝟑 𝟑 𝟑] 𝒂𝒏𝒅 𝑪 = [ 𝟐 − 𝟔 − 𝟐 − 𝟑 𝟓 − 𝟏𝟑 − 𝟒 − 𝟕 −𝟏 𝟒 𝟏 𝟐 𝟎 𝟏 𝟎 𝟏]

Solution

To find the inverse of matrices $B$ and $C$ using the Gauss-Jordan method, we perform row reduction to transform the matrix into its reduced row echelon form (RREF), using the identity matrix. We will apply the Gauss-Jordan elimination process to both matrices one by one.

Matrix $B$ :

0 & 1 & 2 & 2 \\ 1 & 1 & 2 & 3 \\ 2 & 2 & 2 & 3 \\ 2 & 3 & 3 & 3 \end{bmatrix}$$ #### Step-by-Step Process: 1. Augment matrix $$B$$ with the identity matrix of the same size: $$\left[ B \,|\, I \right] = \begin{bmatrix} 0 & 1 & 2 & 2 & 1 & 0 & 0 & 0 \\ 1 & 1 & 2 & 3 & 0 & 1 & 0 & 0 \\ 2 & 2 & 2 & 3 & 0 & 0 & 1 & 0 \\ 2 & 3 & 3 & 3 & 0 & 0 & 0 & 1 \end{bmatrix}$$ 2. Apply row operations to reduce the left side to the identity matrix, while performing the same operations on the right side to obtain the inverse. I will now calculate the inverse of matrix $$B$$ using Gauss-Jordan elimination. #### Performing operations...The inverse of matrix $$B$$ is: $$B^{-1} = \begin{bmatrix} -3 & 3 & -3 & 2 \\ 3 & -4 & 4 & -2 \\ -3 & 4 & -5 & 3 \\ 2 & -2 & 3 & -2 \end{bmatrix}$$ Now, let's proceed with finding the inverse of matrix $$C$$ using the same Gauss-Jordan method. ### Matrix $$C$$: $$C = \begin{bmatrix} 2 & -6 & -2 & -3 \\ 5 & -13 & -4 & -7 \\ -1 & 4 & 1 & 2 \\ 0 & 1 & 0 & 1 \end{bmatrix}$$ I'll calculate the inverse of matrix $$C$$ next.The inverse of matrix $$C$$ is: $$C^{-1} = \begin{bmatrix} -2 & 1 & 0 & 1 \\ 1 & 0 & 2 & -1 \\ -4 & 1 & -3 & 1 \\ -1 & 0 & -2 & 2 \end{bmatrix}$$ Both matrices $$B$$ and $$C$$ now have their inverses calculated using the Gauss-Jordan method. Would you like further details on any steps? ### Related Questions: 1. How can the Gauss-Jordan method be applied to non-square matrices? 2. What are some conditions under which a matrix does not have an inverse? 3. Can you explain the difference between the Gauss-Jordan and Gaussian elimination methods? 4. How is the inverse of a matrix used in solving systems of equations? 5. How can we verify if a matrix inversion result is correct? ### Tip: Always verify the result of a matrix inverse by multiplying the original matrix with the computed inverse. The result should be the identity matrix.

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Math Problem Analysis

Mathematical Concepts

Matrix Inversion
Linear Algebra
Gauss-Jordan Elimination

Formulas

Gauss-Jordan Elimination for Matrix Inversion
Identity Matrix

Theorems

Gauss-Jordan Elimination Theorem

Suitable Grade Level

College Level / Advanced High School (Grades 11-12)

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