Find the LU Decomposition of the following matrix: \[ A = \begin{pmatrix} 2 & 3 \cr 5 & 7 \end{pmatrix} \]

Step 1 ：First, we have to find the lower triangular matrix (L) and the upper triangular matrix (U) such that A = LU.

Step 2 ：The first step is to calculate the first row of U and the first column of L. The element at the first row and first column for U is the same as that of A, i.e., \( u_{11} = a_{11} = 2 \). The element at the first row and first column for L is always 1, i.e., \( l_{11} = 1 \).

Step 3 ：The element at the first row and second column for U is the same as that of A, i.e., \( u_{12} = a_{12} = 3 \). The element at the second row and first column for L is \( a_{21} / u_{11} = 5 / 2 = 2.5 \).

Step 4 ：To calculate the element at the second row and second column for U, we subtract \( l_{21} * u_{12} \) from \( a_{22} \), i.e., \( u_{22} = a_{22} - (l_{21} * u_{12}) = 7 - (2.5 * 3) = -0.5 \). The element at the second row and second column for L is always 1, i.e., \( l_{22} = 1 \).

Step 5 ：Hence, the LU decomposition of A is \[ L = \begin{pmatrix} 1 & 0 \cr 2.5 & 1 \end{pmatrix} \] and \[ U = \begin{pmatrix} 2 & 3 \cr 0 & -0.5 \end{pmatrix} \].