NIMCET 2024
The system of equations , , has infinitely many solutions if
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Answer: Option C —
Detailed Solution
Subtracting the first equation from the second gives ; then , a whole line of solutions.
The third equation must agree with these: , i.e. .
satisfies , so the system then has infinitely many solutions.
✓ Check
With , both and satisfy all three equations.⚠ Common trap
(B) allows , where and the system has no solution.Quick tip
consistent iff ; works
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Matrices — Preparation
This question tests your understanding of Matrices, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.
A matrix is a rectangular arrangement of numbers in rows and columns. This topic covers types of matrices (square, diagonal, scalar, identity, symmetric and skew-symmetric), matrix operations (addition, scalar multiplication and matrix multiplication), the transpose, determinants and their properties, minors and cofactors, the adjoint, and the inverse of a matrix. It also covers solving systems of linear equations using the inverse method and elementary row and column transformations. Remember that matrix multiplication is not commutative, so AB need not equal BA, and that the inverse exists only when the determinant is not zero. Useful results include |AB| = |A||B|, (AB)^-1 = B^-1 A^-1, and |kA| = k^n |A| for a matrix of order n. These properties solve many questions without heavy calculation. Practise the matrices questions below, each with a step-by-step solution.