Matrices & Determinants Question - None

Question

Given system of linear equations:
$x + y + z = 5$ …(i)
$x + 2y + 2z = 6$ …(ii)
$x + 3y + \lambda z = m$ …(iii)
where $\lambda, m \in \mathbb{R}$
If the above system has infinitely many solutions, find $\lambda + m$.

Accepted Answer

Step 1: For infinitely many solutions, the three planes must intersect at a line. This means the third equation must be a linear combination of the first two equations.

Step 2: Express: $(x + 3y + \lambda z - m) = p(x + y + z - 5) + q(x + 2y + 2z - 6)$

Step 3: Expanding the right side:
$(p + q)x + (p + 2q)y + (p + 2q)z - (5p + 6q)$

Step 4: Comparing coefficients:
$p + q = 1$
$p + 2q = 3$
$p + 2q = \lambda$
$5p + 6q = m$

Step 5: Solving: From first two equations, $q = 2$ and $p = -1$

Step 6: Therefore $\lambda = 3$ and $m = 5(-1) + 6(2) = 7$

∴ $\lambda + m = 3 + 7 = \boxed{10}$

Question:

Step-by-Step Solution

Master Matrices & Determinants with Mathbee