Questions: Use the inverse of the coefficient matrix to solve the system of equations. x + 2y + 3z = 28 2x + 3y + 2z = 34 -x - 2y - 4z = -19 (x, y, z) = .

Transcript text: Use the inverse of the coefficient matrix to solve the system of equations. \[ \begin{aligned} x+2 y+3 z & =28 \\ 2 x+3 y+2 z & =34 \\ -x-2 y-4 z & =-19 \end{aligned} \] $(x, y, z)=$ $\square$ . $\square$ $\square$ (Type an ordered triple, using integers or fractions.)

Solution

Solution Steps

Step 1: Construct the Coefficient Matrix $A$

The coefficient matrix $A$ is constructed from the given system of equations. It is a square matrix where each element $a_{ij}$ represents the coefficient of the variable $x_j$ in equation $i$. $$ A = [[ 1, 2, 3], [ 2, 3, 2], [-1, -2, -4]] $$

Step 2: Construct the Constant Vector $B$

The constant vector $B$ contains the constants from the right-hand side of each equation in the system. $$ B = [ 28, 34, -19] $$

Step 3: Calculate the Inverse of the Coefficient Matrix $A^{-1}$

The inverse of matrix $A$ is calculated, which will be used to solve for the vector of variables $x$. $$ A^{-1} = [[-8., 2., -5.], [ 6., -1., 4.], [-1., -0., -1.]] $$

Step 4: Multiply $A^{-1}$ by $B$ to Find the Vector of Variables $x$

The solution vector $x$ is found by multiplying the inverse of the coefficient matrix $A^{-1}$ by the constant vector $B$. $$ x = A^{-1}B = [-61, 58, -9] $$