Test of Rank:
The test of rank determines the consistency of a system of linear equations. If the rank of the coefficient matrix is equal to the rank of the augmented matrix, then the system is consistent. If the rank of the coefficient matrix is greater than the rank of the augmented matrix, then the system is inconsistent.

Given Equations:
2x - y + z = 4
3x - y + z = 6
4x - y + 2z = 7
-x + y - z = 9

Constructing the Coefficient Matrix:
The coefficient matrix is obtained by extracting the coefficients of the variables from the given equations. Let's call this matrix A.

A = | 2 -1 1 |
| 3 -1 1 |
| 4 -1 2 |
|-1 1 -1 |

Constructing the Augmented Matrix:
The augmented matrix is obtained by appending the constants from the right-hand side of the equations to the coefficient matrix. Let's call this matrix B.

B = | 2 -1 1 4 |
| 3 -1 1 6 |
| 4 -1 2 7 |
|-1 1 -1 9 |

Finding the Rank of the Coefficient Matrix:
To find the rank of matrix A, we can perform row operations to reduce it to row-echelon form or reduced row-echelon form. However, we can also use the determinant to find the rank.

The rank of A is equal to the maximum number of linearly independent rows or columns in the matrix. Since A is a 4x3 matrix, the maximum rank of A can be 3.

If the determinant of A is non-zero, then the rank of A is equal to the number of rows or columns, whichever is smaller.

det(A) = | 2 -1 1 |
| 3 -1 1 |
| 4 -1 2 |

Using cofactor expansion along the first row, we have:
det(A) = 2 * | -1 1 |
| -1 2 |

det(A) = 2 * (-2 - (-1))
= 2 * (-2 + 1)
= 2 * (-1)
= -2

Since the determinant of A is non-zero, the rank of A is 3.

Finding the Rank of the Augmented Matrix:
To find the rank of matrix B, we can perform row operations to reduce it to row-echelon form or reduced row-echelon form. However, we can also use the determinant to find the rank.

The rank of B is equal to the maximum number of linearly independent rows or columns in the matrix. Since B is a 4x4 matrix, the maximum rank of B can be 4.

If the determinant of B is non-zero, then the rank of B is equal to the number of rows or columns, whichever is smaller.

det(B) = | 2 -1 1 4 |
| 3 -1 1 6 |
| 4 -1 2 7 |
|-1