Solve by matrix method 2x - 3y 4z = 4 3x y -2z = 9 2x 3y -5z =7?

Question

Solve by matrix method 2x - 3y  4z = 4 3x   y -2z = 9 2x   3y -5z =7?

Answer

Solving a system of linear equations using matrix method

When solving a system of linear equations, one of the methods that can be used is the matrix method. This involves representing the coefficients of the variables and the constants in the system in matrix form, and then using matrix operations to solve for the variables.

Step 1: Write the system of equations in matrix form

The system of equations given is:

2x - 3y + 4z = 4

3x + y - 2z = 9

2x + 3y - 5z = 7

Writing this in matrix form gives:

$$
\begin{bmatrix}
2 & -3 & 4 \
3 & 1 & -2 \
2 & 3 & -5 \
\end{bmatrix}
\begin{bmatrix}
x \
y \
z \
\end{bmatrix}
=
\begin{bmatrix}
4 \
9 \
7 \
\end{bmatrix}
$$

Step 2: Compute the inverse of the coefficient matrix

The next step is to compute the inverse of the coefficient matrix. This is given by:

$$
A^{-1} = \frac{1}{|A|}adj(A)
$$

Where |A| is the determinant of A, and adj(A) is the adjugate matrix of A, which is obtained by taking the transpose of the matrix of cofactors of A.

In this case, the determinant of the coefficient matrix is:

$$
|A| = \begin{vmatrix}
2 & -3 & 4 \
3 & 1 & -2 \
2 & 3 & -5 \
\end{vmatrix}
= 2\begin{vmatrix}
1 & -2 \
3 & -5 \
\end{vmatrix}
-3\begin{vmatrix}
3 & -2 \
2 & -5 \
\end{vmatrix}
+4\begin{vmatrix}
3 & 1 \
2 & 3 \
\end{vmatrix}
= -43
$$

Therefore, the inverse of the coefficient matrix is:

$$
A^{-1} = \frac{1}{-43}
\begin{bmatrix}
-13 & -2 & 5 \
-14 & -2 & 6 \
-3 & -1 & 2 \
\end{bmatrix}
=
\begin{bmatrix}
\frac{13}{43} & \frac{2}{43} & -\frac{5}{43} \
\frac{14}{43} & \frac{2}{43} & -\frac{6}{43} \
\frac{3}{43} & \frac{1}{43} & -\frac{2}{43} \
\end{bmatrix}
$$

Step 3: Compute the solution vector

The solution vector is given by:

$$
\begin{bmatrix}
x \
y \
z \
\end{bmatrix}
=
A^{-1}
\begin{bmatrix}
4 \
9 \
7 \
\end

Answer

2x+y=12 3x-y=13

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