Solve the following system of linear equations using matrix method.
i)2x+y-2z=10
y+10z=-28
3y+16z=-42
To solve the given system of linear equations using the matrix method, we can represent 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 given system of equations can be represented as:
2x + y - 2z = 10 --> Equation 1
0x + y + 10z = -28 --> Equation 2
0x + 3y + 16z = -42 --> Equation 3
Step 1: Create the coefficient matrix [A]
The coefficient matrix [A] consists of the coefficients of the variables x, y, and z from the system of equations.
[A] = | 2 1 -2 |
| 0 1 10 |
| 0 3 16 |
Step 2: Create the variable matrix [X]
The variable matrix [X] consists of the variables x, y, and z.
[X] = | x |
| y |
| z |
Step 3: Create the constant matrix [B]
The constant matrix [B] consists of the constants on the right-hand side of the equations.
[B] = | 10 |
| -28 |
| -42 |
Step 4: Calculate the inverse of the coefficient matrix [A]
To find the inverse of the coefficient matrix [A], we can use various methods such as Gaussian elimination, Gauss-Jordan elimination, or matrix inverse formula. In this case, we'll use the Gauss-Jordan elimination method.
[A | I] = | 2 1 -2 | | 1 0 0 |
| 0 1 10 | | 0 1 0 |
| 0 3 16 | | 0 0 1 |
Perform row operations to transform [A | I] into [I | A^-1]:
| 1 0 0 | | a b c |
| 0 1 0 | | d e f |
| 0 0 1 | | g h i |
Step 5: Multiply the inverse of [A] with the constant matrix [B]
Multiply the inverse of [A] with the constant matrix [B] to obtain the variable matrix [X]:
[X] = [A^-1] * [B]
|x| | a b c | | 10 |
|y| = | d e f | | -28 |
|z| | g h i | | -42 |
Step 6: Solve for x, y, and z
Compute the values of x, y, and z using the obtained variable matrix [X].
x = a*10 + b*(-28) + c*(-42)
y = d*10 + e*(-28) + 42)
z = g*10 + h*(-28) + i*(-42)
By substituting the values of a, b, c, d, e, f, g, h, i from the variable matrix [X], we can solve for x, y, and z.