JNTU B.Tech I Year MATHEMATICS I Previous Question Paper (June - 2022 - R13)

To find the rank of matrix A, we reduce it to row-echelon form using elementary row operations:

1. Perform R2 → R2 - 3R1 and R3 → R3 - R1:
[[1, 3, 4, 3], [0, 0, 0, -6], [0, 0, 0, -2]]
2. Perform R3 → R3 - (1/3)R2:
[[1, 3, 4, 3], [0, 0, 0, -6], [0, 0, 0, 0]]

The number of non-zero rows in the row-echelon form is 2. Therefore, the rank of matrix A is 2.

The system of equations can be represented as an augmented matrix [A|B]:

Applying row operations R2 → R2 - R1 and R3 → R3 - R1:

Applying R3 → R3 - 2R2:

Based on the final row:

1. Unique solution: Occurs when the coefficient matrix is non-singular, i.e., λ - 5 ≠ 0. Thus, λ ≠ 5 and μ can be any value.
2. No solution: Occurs when the rank of the coefficient matrix is less than the rank of the augmented matrix. This happens when λ - 5 = 0 and μ - 8 ≠ 0. Thus, λ = 5 and μ ≠ 8.
3. Infinite solutions: Occurs when the rank of both matrices is equal and less than the number of variables. This happens when λ - 5 = 0 and μ - 8 = 0. Thus, λ = 5 and μ = 8.