Matrices & DeterminantseasyFree

Matrices & Determinants: Equation Pmatrix Pmatrix Pmatrix Pmatrix Pmatrix Pmatrix Sol

JEE Maths question with a full step-by-step solution.

Question

The equation

\begin{pmatrix} 1 & 2 & 2 \\ 1 & 3 & 4 \\ 3 & 4 & k \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}

has a solution

(x, y, z)

besides

(0, 0, 0)

. The value of

k

equals:

0

1

2

correct

3

Solution

Step 1: Condition for a non-trivial solution A homogeneous system has a non-trivial solution if and only if the coefficient determinant is zero:

\begin{vmatrix} 1 & 2 & 2 \\ 1 & 3 & 4 \\ 3 & 4 & k \end{vmatrix} = 0

Step 2: Expand along the first row

1(3k - 16) - 2(k - 12) + 2(4 - 9) = 0

3k - 16 - 2k + 24 - 10 = 0 \implies k - 2 = 0 \implies k = 2

Answer: (3)

Still stuck on this question?Ask your doubt on WhatsApp

Solve more, learn faster

Sign up free to solve more JEE Mathsquestions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.