91Ó°ÊÓ

To write the matrix equation, we need to create a matrix A containing the coefficients of the variables x1, x2, x3, and x4; a matrix x containing the variables; and a matrix b containing the constant terms on the right-hand side. The matrix equation will be of the form: Ax = b. The given system of linear equations is: $$ \begin{aligned} x_{1}+x_{2}+2 x_{3}+3 x_{4}&=4 \\ 2 x_{1}+3 x_{2}-x_{4}&=11 \\ 2 x_{2}-x_{3}+x_{4}&=7 \\ x_{1}+2 x_{2}+x_{3}+x_{4}&=6 \end{aligned} $$ The matrices A, x, and b are: $$ \begin{aligned} A = \begin{pmatrix} 1 & 1 & 2 & 3 \\ 2 & 3 & 0 & -1 \\ 0 & 2 & -1 & 1 \\ 1 & 2 & 1 & 1 \end{pmatrix}, x = \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{pmatrix}, b = \begin{pmatrix} 4 \\ 11 \\ 7 \\ 6 \end{pmatrix} \end{aligned} $$ Therefore, the matrix equation is: $$ \begin{pmatrix} 1 & 1 & 2 & 3 \\ 2 & 3 & 0 & -1 \\ 0 & 2 & -1 & 1 \\ 1 & 2 & 1 & 1 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{pmatrix} = \begin{pmatrix} 4 \\ 11 \\ 7 \\ 6 \end{pmatrix} $$

According to the exercise statement, we should use the inverse of matrix A provided in Exercise 16. Assume we have the inverse: $$ A^{-1} = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} $$ To solve the system using the inverse, we need to multiply both sides of the matrix equation Ax = b by the inverse of A: $$ A^{-1}(Ax) = A^{-1}b $$ Since A is invertible and \(A^{-1}A = I\), where I is the identity matrix, we obtain: $$ x = A^{-1}b $$ Now, we can multiply the inverse of A and the matrix b: $$ \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} \begin{pmatrix} 4 \\ 11 \\ 7 \\ 6 \end{pmatrix} = \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{pmatrix} $$ Finally, the solution to the system of linear equations is represented by the resulting vector x = (x1, x2, x3, x4). Note that the specific values for a_ij are not provided in this exercise, and must be obtained from Exercise 16. If a specific inverse matrix A^{-1} was provided in Exercise 16, plug the given values into the inverse matrix and calculate the multiplication to find the values of x1, x2, x3, and x4.