If \(A = [ * 20 c a&b&c\\ b&c&a\\ c&a&b ],\) where a, b, c are real positive numbers such that abc = 1 and A T A = I then the equation that holds false among the following is

\(A TA = I\) means the rows of A are orthonormal: Each row has length 1: \(a 2 + b 2 + c 2 = 1\) Different rows have dot product 0: \(ab + bc + ca = 0\) Then \((a+b+c) 2 = 1 + 2(0) = 1\), so \(a + b + c = 1\) (taking the positive value). \(a 3 + b 3 + c 3 = 3abc + (a+b+c)[(a 2+b 2+c 2) - (ab+bc+ca)] = 3 + 1 × 1 = 4\) So (A), (B) and (D) hold, and (C) \(ab + bc + ca = 1\) is false (it equals 0)....

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Question 2

Mathematics Determinants Hard

If \(A = \left[ {\begin{array}{*{20}{c}} a&b&c\\ b&c&a\\ c&a&b \end{array}} \right],\) where a, b, c are real positive numbers such that abc  = 1 and ATA = I then the equation that holds false among the following is

(A) a + b + c = 1

(B) a2 + b2 + c2 = 1

(C) ab + bc + ca = 1

(D) a3 + b3 + c3 = 4

✓ Correct Answer: Option C Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

\(A^TA = I\) means the rows of A are orthonormal:

Each row has length 1: \(a^2 + b^2 + c^2 = 1\)
Different rows have dot product 0: \(ab + bc + ca = 0\)

Then \((a+b+c)^2 = 1 + 2(0) = 1\), so \(a + b + c = 1\) (taking the positive value).

\(a^3 + b^3 + c^3 = 3abc + (a+b+c)\left[(a^2+b^2+c^2) - (ab+bc+ca)\right] = 3 + 1 \times 1 = 4\)

So (A), (B) and (D) hold, and (C) \(ab + bc + ca = 1\) is false (it equals 0).

Answer: (C)

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Question 2

📘 Step-by-Step Solution & Detailed Explanation:

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