3. please give the Matrix Equation for the following equations \( \left\{\begin{array}{l}x+2 y+3 z=1 \\ 2 x+3 y+4 z=2 \\ 3 x+4 y+5 z=3\end{array}\right. \)

To express the given system of equations in matrix form, we can represent it as \( A \mathbf{x} = \mathbf{b} \), where: - \( A \) is the coefficient matrix, - \( \mathbf{x} \) is the vector of variables, and - \( \mathbf{b} \) is the vector of constants. Given the system of equations: \[ \begin{align*} 1. & \quad x + 2y + 3z = 1 \\ 2. & \quad 2x + 3y + 4z = 2 \\ 3. & \quad 3x + 4y + 5z = 3 \end{align*} \] We can identify the coefficients of \( x \), \( y \), and \( z \) for each equation: - For the first equation: coefficients are \( [1, 2, 3] \) - For the second equation: coefficients are \( [2, 3, 4] \) - For the third equation: coefficients are \( [3, 4, 5] \) Thus, the coefficient matrix \( A \) is: \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix} \] The vector of variables \( \mathbf{x} \) is: \[ \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \] The vector of constants \( \mathbf{b} \) is: \[ \mathbf{b} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \] Putting it all together, the matrix equation can be written as: \[ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \] This is the matrix equation corresponding to the given system of equations.