Concept 09 / 09Matrices and Determinants
Homogeneous Systems and Applications: Non-Trivial Solutions, Concurrency and Coplanarity
Why AX = O is always consistent, the determinant condition for non-trivial solutions, eliminating variables to get conditions on parameters, and determinant conditions for concurrent lines, collinear points and the consistency of three equations in two unknowns.
A system is homogeneous when every constant term is zero: . It always has the trivial solution , so it is always consistent. The only question is whether it has other solutions.
If , then is the only solution. If , there are infinitely many solutions, including non-trivial ones. The reason is dimensional: means the columns of are linearly dependent, so some non-zero combination of them is zero, and the coefficients of that combination form a non-trivial solution. Any scalar multiple of a solution is again a solution, so there are infinitely many. So:
has a non-trivial solution .
This condition is the most used determinant fact in JEE algebra. It turns 'there exist , not all zero, such that...' into a single equation.
Finding the solutions when (for ). If two of the equations are independent, solve them for two variables in terms of the third, or use the cross-multiplication rule: from and ,
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Geometrically, is parallel to the cross product of the two coefficient vectors, since it is perpendicular to both.
Elimination. Eliminating variables from a set of linear equations produces a determinant condition. For instance, to eliminate from , , (with not all zero), rewrite as a homogeneous system and set its determinant to zero: , which simplifies to .
Applications to coordinate geometry.
Concurrency of three lines. The lines , , pass through a common point only if . Reason: a common point gives a non-trivial solution of the homogeneous system. The converse needs care: the determinant also vanishes when the three lines are parallel (they 'meet at infinity'), so check that no two are parallel.
Consistency of three equations in two unknowns. The same determinant condition, with the constants as the third column, is necessary for () to have a common solution.
Collinearity of three points: the area determinant with a column of s is zero.
Coplanarity in three dimensions: four points are coplanar exactly when the determinant with rows is zero, equivalently the scalar triple product of three edge vectors is zero.
Key Takeaways:
- is always consistent; is always a solution.
- Non-trivial solutions exist , and then there are infinitely many.
- Two independent equations: .
- 'Not all zero' in a question signals the condition .
- Three lines concurrent determinant of coefficients ; check they are not all parallel.
- Three equations in two unknowns consistent the augmented determinant is .
- Pitfall: treating as 'no solution' for a homogeneous system. It means infinitely many.
Brain-Teaser 1: Find for which , , has a non-trivial solution.
Solution: Rearrange to , , . Set the determinant to zero. Every row sums to , so add all columns into the first: the determinant becomes . Subtract row 1 from rows 2 and 3: and . The determinant is , whose discriminant is . So the only real value is . (In the language of the next concept, is an eigenvalue of the coefficient matrix, and is its only real eigenvalue.) At , solves all three equations, confirming a non-trivial solution.
Brain-Teaser 2: For what are the lines , and concurrent?
Solution: The lines , and are concurrent only if . Expand along the first row: . So . Check: the first two lines meet at , and . Since the first two lines are not parallel, the determinant condition is sufficient here.
Brain-Teaser 3: If the equations , , have a non-trivial solution, show that (assume ).
Solution: Add to both sides of the first equation: , so where . Similarly and . For a non-trivial solution, : if , each equation would give (from and so on). Adding the three, . Since , this reads , so . A clever rearrangement can sometimes replace the determinant condition entirely, but it still encodes the same fact: a non-trivial solution exists only when the equations are dependent.
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