1. **Stating the problem:** We are given matrices \(A = \begin{pmatrix} 2 & 1 \ \end{pmatrix}\) and \(B = \begin{pmatrix} 5 & 7 \end{pmatrix}\). We need to find matrix \(X\) such that \(AX = B\). 2. **Understanding the problem:** Matrix \(A\) is a 1x2 matrix and \(B\) is a 1x2 matrix. The equation \(AX = B\) implies multiplication of \(A\) (1x2) by \(X\) must result in \(B\) (1x2). 3. **Matrix dimensions:** Let \(X\) be a 2x2 matrix \(X = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix}\). Then \(AX = \begin{pmatrix} 2 & 1 \end{pmatrix} \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} = \begin{pmatrix} 2x_{11} + 1x_{21} & 2x_{12} + 1x_{22} \end{pmatrix}\). 4. **Set equal to B:** \[ \begin{pmatrix} 2x_{11} + x_{21} & 2x_{12} + x_{22} \end{pmatrix} = \begin{pmatrix} 5 & 7 \end{pmatrix} \] 5. **Solve for elements of X:** We have two equations: \[ 2x_{11} + x_{21} = 5 \] \[ 2x_{12} + x_{22} = 7 \] 6. **Infinite solutions:** Since there are 4 unknowns and only 2 equations, the system is underdetermined. We can express \(x_{21}\) and \(x_{22}\) in terms of \(x_{11}\) and \(x_{12}\): \[ x_{21} = 5 - 2x_{11} \] \[ x_{22} = 7 - 2x_{12} \] 7. **General solution:** \[ X = \begin{pmatrix} x_{11} & x_{12} \\ 5 - 2x_{11} & 7 - 2x_{12} \end{pmatrix} \] where \(x_{11}\) and \(x_{12}\) are arbitrary real numbers. **Final answer:** The matrix \(X\) satisfying \(AX = B\) is $$ X = \begin{pmatrix} x_{11} & x_{12} \\ 5 - 2x_{11} & 7 - 2x_{12} \end{pmatrix} $$ with \(x_{11}, x_{12} \in \mathbb{R}\).