Solve the following simultaneous equations using matrices method. x+2y+3z=5, x-y-z=0, 4x+5y+6z=11

Question

ScanSolve AI Tutor · Accepted Answer

Okay, Bashiru, let's solve question 4a using the matrices method.

The given system of simultaneous equations is:

$x + 2y + 3z = 5$
$x - y - z = 0$
$4x + 5y + 6z = 11$

Write the system in matrix form $AX = B$ .
$A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & -1 & -1 \\ 4 & 5 & 6 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 5 \\ 0 \\ 11 \end{bmatrix}$
Calculate the determinant of matrix A.
$\det(A) = 1 \begin{vmatrix} -1 & -1 \\ 5 & 6 \end{vmatrix} - 2 \begin{vmatrix} 1 & -1 \\ 4 & 6 \end{vmatrix} + 3 \begin{vmatrix} 1 & -1 \\ 4 & 5 \end{vmatrix}$ $\det(A) = 1((-1)(6) - (-1)(5)) - 2((1)(6) - (-1)(4)) + 3((1)(5) - (-1)(4))$ $\det(A) = 1(-6 + 5) - 2(6 + 4) + 3(5 + 4)$ $\det(A) = 1(-1) - 2(10) + 3(9)$ $\det(A) = -1 - 20 + 27$ $\det(A) = 6$
Find the adjoint of matrix A.
First, find the cofactor matrix $C$ :
$C_{11} = -1, \quad C_{12} = -10, \quad C_{13} = 9$ $C_{21} = 3, \quad C_{22} = -6, \quad C_{23} = 3$ $C_{31} = 1, \quad C_{32} = 4, \quad C_{33} = -3$
So, the cofactor matrix is:
$C = \begin{bmatrix} -1 & -10 & 9 \\ 3 & -6 & 3 \\ 1 & 4 & -3 \end{bmatrix}$
The adjoint matrix is the transpose of the cofactor matrix:
$\text{adj}(A) = C^T = \begin{bmatrix} -1 & 3 & 1 \\ -10 & -6 & 4 \\ 9 & 3 & -3 \end{bmatrix}$
Calculate the inverse of matrix A.
$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$ $A^{-1} = \frac{1}{6} \begin{bmatrix} -1 & 3 & 1 \\ -10 & -6 & 4 \\ 9 & 3 & -3 \end{bmatrix}$
Calculate $X = A^{-1}B$ .
$X = \frac{1}{6} \begin{bmatrix} -1 & 3 & 1 \\ -10 & -6 & 4 \\ 9 & 3 & -3 \end{bmatrix} \begin{bmatrix} 5 \\ 0 \\ 11 \end{bmatrix}$ $X = \frac{1}{6} \begin{bmatrix} (-1)(5) + (3)(0) + (1)(11) \\ (-10)(5) + (-6)(0) + (4)(11) \\ (9)(5) + (3)(0) + (-3)(11) \end{bmatrix}$ $X = \frac{1}{6} \begin{bmatrix} -5 + 0 + 11 \\ -50 + 0 + 44 \\ 45 + 0 - 33 \end{bmatrix}$ $X = \frac{1}{6} \begin{bmatrix} 6 \\ -6 \\ 12 \end{bmatrix}$ $X = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}$
Therefore, the solution is:
$x = \boxed{1}$ $y = \boxed{-1}$ $z = \boxed{2}$

Solve the following simultaneous equations using matrices method. x+2y+3z=5, x−y−z=0, 4x+5y+6z=11

Answer

Step-by-step solution

Write the system in matrix form $AX = B$ .

Calculate the determinant of matrix A.

Find the adjoint of matrix A.

Calculate the inverse of matrix A.

Calculate $X = A^{-1}B$ .

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