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Matrices & Determinants
System of linear equations
Grade Class 12

Question:

The number of real values &lambda;, such that the system of linear equations<br>2x - 3y + 5z = 9<br>x + 3y - z = -18<br>3x - y + (&lambda;<sup>2</sup> - |&lambda;|)z = 16<br>has no solution, is :-
(1) 0
(2) 1
(3) 2
(4) 4

Step-by-Step Solution

Key Concept: A system of linear equations has no solution if the determinant of the coefficient matrix is zero and the system is inconsistent (i.e., the augmented matrix has a higher rank than the coefficient matrix).
The system is given by:<br>2x - 3y + 5z = 9<br>x + 3y - z = -18<br>3x - y + (&lambda;<sup>2</sup> - |&lambda;|)z = 16<br>The determinant of the coefficient matrix D = |2 -3 5; 1 3 -1; 3 -1 &lambda;<sup>2</sup>-|&lambda;|| = 2(3(&lambda;<sup>2</sup>-|&lambda;|) - 1) + 3((&lambda;<sup>2</sup>-|&lambda;|) + 3) + 5(-1 - 9) = 6(&lambda;<sup>2</sup>-|&lambda;|) - 2 + 3(&lambda;<sup>2</sup>-|&lambda;|) + 9 - 50 = 9(&lambda;<sup>2</sup>-|&lambda;|) - 43.<br>For no solution, D = 0, so 9(&lambda;<sup>2</sup>-|&lambda;|) = 43. This quadratic in |&lambda;| gives two distinct positive values for |&lambda;|, leading to two real values for &lambda;.
Correct Answer: 3

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