This category gathers questions focused on the fundamentals of matrices and determinants. It covers matrix operations, properties, and determinant calculations, helping students build a solid base for further linear algebra studies.
In easy words, we verify 0 by full calculation: use standard identity, substitute, simplify step by step to reach 0.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
Let's break it down: complete solution: write formula, plug data, do arithmetic. Final value matches infinitely many solutions.. Simple and accurate.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
From the definition, after simplifying, -2 is left. The method is simple: formula → substitution → calculation → answer.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
In easy words, for homogeneous infinite solutions det must be zero. Det=1(-α-3)-1(α-2)+1(3+2)=-2α+4=0, so α=2. Solved completely.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
Look at it this way: rule: unique iff |A|≠0. If |A|=0 and (adjA)B=0, you get infinite solutions. So i and iii correct. Full logic.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
From the definition, for unique trivial solution need |A|≠0. Matrix [[2,-3],[2,α]], det=2α+6≠0, so α≠-3. Complete determinant check.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
Applying the basic concept, homogeneous AX=0 always has trivial solution X=0. So I is true, II about non-homogeneous may fail. Complete reasoning.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
Let's break it down: formula |kA|=kⁿ|A|, n=order. Here n=2, k=5, |A|=5. So |5A|=5²×5=25×5=125. Full calculation.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
We know that area =½|det| where det=|x₁ y₁ 1;x₂ y₂ 1;x₃ y₃ 1|. Equilateral area=√3/4 a². So ½|det|=√3/4 a², det²=3a⁴/4.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
Here's the idea in simple terms: formula det(kA)=k³detA for 3×3. So det(2AB)=2³·detA·detB=8×3×4=96. Complete calculation step by step.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
In easy words, after simplifying, 0 is left. The method is simple: formula → substitution → calculation → answer.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
From the definition, A=kI₃. Inverse formula (kI)⁻¹=(1/k)I. Because kI×(1/k)I=I₃. So answer (1/k)I. Full solution.
Ref: NCERT Mathematics Class XII — Matrices and Determinants
