Tamil Nadu Board of Secondary EducationHSC Arts Class 11th

# Consider the matrix Aα = [cosα-sinαsinαcosα] Find all possible real values of α satisfying the condition AATAα+AαT = I - Mathematics

Sum

Consider the matrix Aα = [(cos alpha, - sin alpha),(sin alpha, cos alpha)] Find all possible real values of α satisfying the condition "A"_alpha + "A"_alpha^"T" = I

#### Solution

Aα = [(cos alpha, - sin alpha),(sin alpha, cos alpha)]

"A"_alpha^"T" = [(cos alpha, sin alpha),(-sin alpha, cos alpha)]

"A"_alpha + "A"_alpha^"T" = [(cos alpha, - sin alpha),(sin alpha, cos alpha)] + [(cos alpha, sin alpha),(-sin alpha, cos alpha)]

= [(cos alpha + cos alpha, -sinalpha + sin alpha),(sin alpha - sin alpha, cos alpha + cos alpha)]

"A"_alpha + "A"_alpha^"T" = [(2cos alpha, 0),(0, 2 cos alpha)]

Given "A"_alpha + "A"_alpha^"T" = I

∴ [(2cos alpha, 0),(0, 2 cos alpha)] = [(1, 0),(0, 1)]

2 cos  alpha = 1

⇒ cos allpha = 1/2

The general solution is α = 2"n"  pi +-  pi/3, "n" ∈ "Z"

Concept: Matrices
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#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 11th Mathematics Volume 1 and 2 Answers Guide
Chapter 7 Matrices and Determinants
Exercise 7.1 | Q 6. (ii) | Page 18
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