Question
Find the inverse of the matrix
using row reduction.
Answer :
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Skip in 10
To find the inverse of the given matrix using row reduction, we'll follow these steps: 1. Write the augmented matrix \([A | I]\), where \(A\) is the given matrix and \(I\) is the identity matrix of the same size. 2. Perform row operations to transform the left side of the augmented matrix into the identity matrix \(I\). 3. The right side of the augmented matrix will then be the inverse of \(A\), denoted as \(A^{-1}\). Let's proceed step by step. --- ### Step 1: Write the augmented matrix \([A | I]\) The given matrix \(A\) is: \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix} \] The identity matrix \(I\) of the same size is: \[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] The augmented matrix \([A | I]\) is: \[ [A | I] = \begin{bmatrix} 1 & 2 & 3 & | & 1 & 0 & 0 \\ 0 & 1 & 4 & | & 0 & 1 & 0 \\ 5 & 6 & 0 & | & 0 & 0 & 1 \end{bmatrix} \] --- ### Step 2: Perform row operations #### Goal: Transform the left side of \([A | I]\) into \(I\). 1. Row 1: Already has a leading 1 in the first column. No action needed. 2. Row 3: Subtract \(5 \times \text{Row 1}\) from Row 3 to eliminate the \(5\) in the first column: \[ R_3 = R_3 - 5R_1 \] \[ \begin{bmatrix} 5 & 6 & 0 \end{bmatrix} - 5 \times \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 0 & -4 & -15 \end{bmatrix} \] The augmented matrix becomes: \[ \begin{bmatrix} 1 & 2 & 3 & | & 1 & 0 & 0 \\ 0 & 1 & 4 & | & 0 & 1 & 0 \\ 0 & -4 &
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