Find the LU Decomposition of the matrix \( A = \begin{bmatrix} 4 & 3 \newline 6 & 3 \end{bmatrix} \)
Answer
Step 3: Inverse the operations performed on \( A \) to get \( L \). Divide row 2 by \( \frac{4}{6} \) and add row 1 to it. Now, \( L = \begin{bmatrix} 1 & 0 \newline \frac{3}{2} & 1 \end{bmatrix} \)
Step 1 ：Step 1: Initialize the identity matrix \( I = \begin{bmatrix} 1 & 0 \newline 0 & 1 \end{bmatrix} \) and the matrix \( L = I \)
Step 2 ：Step 2: Perform row operations to convert \( A \) into an upper triangular matrix \( U \). Multiply row 2 by \( \frac{4}{6} \) and subtract row 1 from it. Now, \( U = \begin{bmatrix} 4 & 3 \newline 0 & -1 \end{bmatrix} \)
Step 3 ：Step 3: Inverse the operations performed on \( A \) to get \( L \). Divide row 2 by \( \frac{4}{6} \) and add row 1 to it. Now, \( L = \begin{bmatrix} 1 & 0 \newline \frac{3}{2} & 1 \end{bmatrix} \)
