Question

Solve by matrix method:2x+y-z=-7,3x-y+2z=18 and x+2y+3z=13

Answer 1

To solve the system of equations using the matrix method, we can write the system in matrix form:

[A][X] = [B]

where [A] is the coefficient matrix, [X] is the variable matrix, and [B] is the constant matrix.

The coefficient matrix [A] is:

| 2 1 -1 |
| 3 -1 2 |
| 1 2 3 |

The variable matrix [X] is:

| x |
| y |
| z |

The constant matrix [B] is:

| -7 |
| 18 |
| 13 |

We can then solve for [X] by taking the inverse of [A] and multiplying it by [B]:

[X] = [A]^-1 * [B]

First, let's find the inverse of [A]. Using matrix methods or Gaussian elimination, we find that the inverse of [A] is:

| -0.538 0.231 0.462 |
| 0.615 0.154 -0.385 |
| -0.231 -0.077 0.308 |

Next, we multiply the inverse of [A] by [B]:

| x | | -0.538 0.231 0.462 | | -7 |
| y | = | 0.615 0.154 -0.385 | * | 18 |
| z | | -0.231 -0.077 0.308 | | 13 |

Calculating the products:

| x | | -0.538 0.231 0.462 | | -7 | | -4.923 |
| y | = | 0.615 0.154 -0.385 | * | 18 | = | -8.308 |
| z | | -0.231 -0.077 0.308 | | 13 | | 3.077 |

Therefore, the solution to the system of equations is:

x = -4.923
y = -8.308
z = 3.077