Complete, verified solutions to the NCERT Class 12 Maths Chapter 3 (Matrices) exercise questions. These Class 12 Mathematics Chapter 3 solutions are also useful as quick revision notes before exams.
NCERT Exercise Solutions
3.1 In the matrix A = [[2,5,19,-7],[35,-2,5/2,12],[√3,1,-5,17]], write (i) the order, (ii) the number of elements, (iii) elements a₂₁, a₃₃, a₂₃.
Ans: (i) Order 3×4. (ii) 12 elements. (iii) a₂₁=35, a₃₃=−5, a₂₃=5/2.
3.2 How many possible orders are there for a matrix with 24 elements? With 13 elements?
Ans: 24 = 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1 ⇒ 8 possible orders. For 13 (prime): 1×13, 13×1 ⇒ 2 possible orders.
3.3 Construct a 2×2 matrix A=[aᵢᵇ] whose elements are given by aᵢᵇ=(i+j)²/2.
Ans: a₁₁=(1+1)²/2=2, a₁₂=(1+2)²/2=4.5, a₂₁=(2+1)²/2=4.5, a₂₂=(2+2)²/2=8. So A=[[2,4.5],[4.5,8]].
3.4 If A=[[1,1],[-1,1]], B=[[a,-a],[b,b]], find a,b such that (A+B)²=A²+B².
Ans: Expanding (A+B)²=A²+AB+BA+B², this equals A²+B² only when AB+BA=0 (the zero matrix), not when AB=BA. Computing AB=[[a+b,b-a],[b-a,a+b]] and BA=[[2a,0],[0,2b]], so AB+BA=[[3a+b,b-a],[b-a,a+3b]]. Setting every entry to zero: b-a=0 gives b=a, and substituting into 3a+b=0 gives 4a=0, so a=0 and b=0.
3.5 (Express as sum of symmetric/skew-symmetric) For A=[[3,5],[1,-1]], express A as P+Q where P is symmetric and Q is skew-symmetric.
Ans: P=(A+Aᵀ)/2=[[3,3],[3,-1]], Q=(A-Aᵀ)/2=[[0,2],[-2,0]]. Check: P+Q=[[3,5],[1,-1]]=A. ✓
Class 12 Mathematics Chapter 3 – Notes and Extra Questions
Along with these NCERT Solutions, students can also use the Class 12 Mathematics Chapter 3 Extra Questions and Class 12 Mathematics Chapter 3 Revision Notes for quick revision and extra practice.
