Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 14, 2023

Solve the system using the given inverse coefficient matrix: $x + 2y + 3z = -3$ , $x + y + z = 4$ , $-x + y + 2z = -2$ .

STEP 1

Assumptions1. We have a system of linear equations in the form of Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the vector of constants. . The inverse of the coefficient matrix A is given.

STEP 2

We can write the system of equations in matrix form. The coefficient matrix A is $A = \begin{bmatrix} 1 &2 & \\ 1 &1 &1 \\ -1 &1 &2\end{bmatrix}$ The vector of variables x is $x = \begin{bmatrix} x \\ y \\ z\end{bmatrix}$ And the vector of constants b is $b = \begin{bmatrix} - \\ 4 \\ -2\end{bmatrix}$ So, the system of equations can be written as $Ax = b$

STEP 3

We can solve for x by multiplying both sides of the equation by the inverse of A. The inverse of A is given as $A^{-1} = \begin{bmatrix} 1 & -1 & -1 \\ -3 &5 &2 \\ 2 & -3 & -1\end{bmatrix}$ So, we have $A^{-1}Ax = A^{-1}b$

STEP 4

The product of a matrix and its inverse is the identity matrix, so we have $Ix = A^{-1}b$

STEP 5

The identity matrix I times any vector is just the vector itself, so we have $x = A^{-1}b$

STEP 6

Now we can plug in the given values for A^{-1} and b to solve for x $x = \begin{bmatrix} 1 & -1 & -1 \\ -3 &5 &2 \\ 2 & -3 & -1\end{bmatrix} \begin{bmatrix} -3 \\ 4 \\ -2\end{bmatrix}$

STEP 7

Perform the matrix multiplication to find the solution vector x $x = \begin{bmatrix} 1(-3) -14 -1(-2) \\ -3(-3) +54 +2(-2) \\ 2(-3) -34 -1*(-2) \end{bmatrix} = \begin{bmatrix} -1 \\ 23 \\ -14\end{bmatrix}$ So, the solution to the system of equations is x = -1, y =23, z = -14.

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