Question

Consider the linear system of non homogeneous equation in three variables AX = B. $\rho(A)$ and $\rho([A:B])$ are roots of the equation $x^2+bx+c=0$. AX = B has infinite number of solutions, then the option containing two possible pairs of values of (b,c) is (Note: $\rho$ denotes rank.)

• A. (1,1) (2,1)
• B. (1,--2) (4,--4)
• C. (--4,4) (--2,1)
• D. (1,1) (1,2)

Hint:

• For infinite solutions to \(AX=B\) (n variables): \(\rho(A) = \rho([A:B]) = r<n\).
• If \(r_1, r_2\) are roots of \(x^2+bx+c=0\), and \(r_1=r_2=r\) for infinite solutions, then the quadratic must have equal roots.
• Condition for equal roots: Discriminant \(D = b^2-4c = 0 \implies b^2=4c\).
• The equal root is \(x = -b/2\). So, \(r = -b/2\).
• Since \(r\) is a rank, \(r\) must be a non-negative integer. For this problem, \(r=1\) or \(r=2\) as \(r<3\).

Answer 1

The Correct Option is C

The given problem involves analyzing the properties of a linear system of equations with the condition that it has an infinite number of solutions. We consider a non-homogeneous linear system in matrix form as AX = B, where A is the coefficient matrix and B is the constant matrix. For such a system to have infinite solutions, the rank conditions must satisfy: $\rho(A) = \rho([A:B]) < n$, where n is the number of unknowns (in this case, 3).

Given that the ranks of $\rho(A)$ and $\rho([A:B])$ are roots of the quadratic equation $x^2+bx+c=0$, and $\rho(A) = \rho([A:B])$, the roots must be equal.

Thus, for $\rho(A) = \rho([A:B]) = r$, $x^2+bx+c=0$ becomes $(x-r)^2=0$. This implies a double root, meaning:

• $b = -2r$
• $c = r^2$

The condition for infinite solutions is satisfied when $r = 2$ because $r=3$ would mean full rank, allowing a unique solution only. Consequently, we have:

• For $r=2$: $b = -4$ and $c = 4$
• For $r=1$: $b = -2$ and $c = 1$

Therefore, the possible pairs of ($b$, $c$) are (-4, 4) and (-2, 1).

Correct Answer: The correct options matching these pairs are (--4,4) and (--2,1).