solve the equations by using
a) the adjoint matrix
b) elementary row operations
i) x+y+z=6
2x-y+z=3
x+4y-z=6
ii) 3p-3q+r=1
-3p+5q-2r=-1
p-2q+r=3
iii)-12a+b+17c=-14
2a-b+3c=5
8a+b-13c=-3
thanks sir/madam
X=3.Y=5
a) To solve the equations using the adjoint matrix, we need to follow these steps:
Step 1: Set up the coefficient matrix A and the constant vector B based on the given system of equations. For example, for the first system of equations:
A =
| 1 1 1 |
| 2 -1 1 |
| 1 4 -1 |
B =
| 6 |
| 3 |
| 6 |
Step 2: Calculate the determinant of A. If the determinant is non-zero, we can proceed to the next step. Otherwise, the system is either inconsistent or has infinitely many solutions.
Step 3: Calculate the adjoint of the matrix A, denoted by adj(A). The adjoint matrix is obtained by taking the transpose of the cofactor matrix.
Step 4: Use the formula X = adj(A) * B to find the solution vector X.
For the given system i), you can apply the above steps to find the solution using the adjoint matrix.
b) To solve the equations using elementary row operations, we need to follow these steps:
Step 1: Set up the augmented matrix by combining the coefficient matrix A and the constant vector B. For example, for the first system of equations:
[ A | B ] =
| 1 1 1 | 6 |
| 2 -1 1 | 3 |
| 1 4 -1 | 6 |
Step 2: Use elementary row operations to transform the augmented matrix into row-echelon form or reduced row-echelon form. These operations include swapping rows, multiplying a row by a scalar, and adding or subtracting rows.
Step 3: Once the augmented matrix is in row-echelon or reduced row-echelon form, read off the solution directly from the matrix.
Repeat the above steps for the given systems ii) and iii) to solve the equations using elementary row operations.