Matrices & Determinants Question - None

Question

Given system of linear equations:

$x + y + z = 6$ ... (i)

$4x + \lambda y - \lambda z = \lambda - 2$ ... (ii)

$3x + 2y - 4z = -5$ ... (iii)

has infinitely many solutions. Find the value of $\lambda$.

Accepted Answer

Step 1: For a system to have infinitely many solutions, the determinant of the coefficient matrix must be zero: $D = 0$

Step 2: Calculate the determinant:

$$\begin{vmatrix} 1 & 1 & 1 \ 4 & \lambda & -\lambda \ 3 & 2 & -4 \end{vmatrix} = 0$$

Step 3: Expand along the first row:

$1(-4\lambda + 2\lambda) - 1(-16 + 3\lambda) + 1(8 - 3\lambda) = 0$

Step 4: Simplify:

$-8\lambda + 2\lambda + 16 - 3\lambda + 8 - 3\lambda = 0$

$-8\lambda + 24 = 0$

$\lambda = 3$

∴ Answer is $\lambda = 3$

Question: