Hindi
English

Question

Download Solution PDF

मान लीजिए A एक 2 × 2 वास्तविक आव्यूह है जिसमें det A = 1 और ट्रेस A = 3 है। ट्रेस A2 का मान क्या है?

This question was previously asked in
CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
Download PDF Attempt in App
View all CSIR NET Papers >
  1. 2
  2. 10
  3. 9
  4. 7

Answer (Detailed Solution Below)

Option 4 : 7
Free Tests
View all Free tests >
Free
Seating Arrangement
10 Qs. 20 Marks 15 Mins
Start Now

Detailed Solution

Download Solution PDF

गणना:

मान लीजिए a और b, A के दो आइगेन मान हैं।

a + b = 3 . . . . . . 1

ab = 1 . . . . . . . 2

(a - b )2 = (a +b)2 - 4ab

(a-b ) 2 = 5

a - b = . . . . 3

अब, समीकरण 1 और 3 से, हमें मिलता है

a = और b =

trace ( A2 ) = a2 + b2 = 7

इसलिए विकल्प (4) सही है।

Alternate Method 

विधि -I मान लीजिए λ1 और λ2 A के दो आइगेन मान हैं।

तब det A = λ1 λ2 = 1

& Trace A = λ1 + λ2 = 3

और, A2 के आइगेन मान λ12, λ22 हैं।

⇒ Trace (A2) = λ12 + λ22

अब, (λ1 + λ2)2 = λ12 + λ22 + 2λ1λ2

⇒ (3)2 = λ12 + λ22 + 2

⇒ λ12 + λ22 = 9 - 2 = 7

इसलिए विकल्प (4) सही है।

विधि - II

ChA (x) = x2 - Trace (A) x + Det (A) = 0

⇒ x2 - 3x + 1 = 0 ⇒

λ2 जो A के आइगेन मान हैं।

तब A2 के आइगेन मान हैं,

तब trace = 7

इसलिए विकल्प (4) सही है।

Download Solution PDF
Latest CSIR NET Updates

Last updated on Jul 19, 2025

-> The CSIR NET June 2025 Exam Schedule has been released on its official website. The exam will be held on 28th July 2025.

-> CSIR NET City Intimation Slip 2025 has been released @csirnet.nta.ac.in.

-> The CSIR UGC NET is conducted in five subjects -Chemical Sciences, Earth Sciences, Life Sciences, Mathematical Sciences, and Physical Sciences. 

-> Postgraduates in the relevant streams can apply for this exam.

-> Candidates must download and practice questions from the CSIR NET Previous year papers. Attempting the CSIR NET mock tests are also very helpful in preparation.

‹‹ Previous Ques
Next Ques ››
Exams Tests Super Practice

More Algebra of Linear Transformations Questions

Q1.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q2.मानें कि A एक 3 × 3 वास्तविक प्रविष्टियों वाला आव्यूह है जिसका अभिलक्षणिक बहुपद p(T) है जो T2 से भाज्य है। निम्न वक्तव्यों में से कौन सा सत्य है?
Q3.यदि A को 2 × 2 वास्तविक आव्यूह मानें जिसके लिए detA = 1 तथा अनुरेख (ट्रेस) A = 3 हो तो अनुरेख (ट्रेस) A2 का मान क्या है?
Q4.यदि \(A=\left(\begin{array}{ll}3 & -2 \\ 2 & -1\end{array}\right)\), तब A20 के बराबर है
Q5.A तथा B को 2 × 2 के आव्यूह मानें। तब निम्न में से कौन-से सत्य है?
Q6.मानें कि A एक n × n आव्यूह है जिसके सभी शून्येतर अभिलक्षणिक मानों के समुच्चय में r अवयव हैं। तब निम्न में से कौन-सा सत्य है?

More Linear Algebra Questions

Q1.Let A be a 10 x 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true? 
Q2.For a quadratic form f(x, y, z) ∈ ℝ[x, y z], we say that (a, b, c) ∈ ℝ3 is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)? 
Q3.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q4.Let V be the real vector space of 2 x 2 matrices with entries in ℝ. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where \(\rm A=\begin{pmatrix}2&0\\\ 0&1\end{pmatrix}\). What is the characteristic polynomial of T? 
Q5.Let \(\rm A = \begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&1\\\ 0&0&1&1\end{pmatrix}\) and consider the symmetric bilinear form on R4 given by (v, w) = vt Aw, for v, w ∈ ℝ4. Which of the following statements is true? 
Q6.माना \(\rm M=\begin{bmatrix}4&-3\\\ 1&0\end{bmatrix}\) है, निम्नलिखित कथनों पर विचार कीजिए: 𝑃: 𝑀8 + 𝑀12 विकर्णीय है। 𝑄: 𝑀7 + 𝑀9 विकर्णीय है। निम्नलिखित कथनों में से कौन सा सही है?
Q7.नियतांकों a तथा b पर इस प्रकार विचार कीजिए कि (p, q) से (P, Q) पर निम्न प्रसामान्यीकृत निर्देशांक रूपांतरण विहित है Q = pq(a+1), P = qb a तथा b के मान क्या हैं?
Q8.निम्न आव्यूह से सहचारी द्विघात रूप (quadratic form) Q(x, y, z) पर विचार करें B = \(\left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & -2 \end{array}\right]\) मानें कि S = \(\left\{\left[\begin{array}{l} \rm a \\\rm b \\\rm c\end{array}\right] \in \rm ℝ^3 \mid Q(a, b, c)=0\right\}\) निम्न वक्तव्यों में से कौन-सा असत्य है?
Q9.मानें कि किसी निश्चित n ≥ 2 के लिए ℝn में x = (x1, …, xn) तथा y = (y1, …, yn) दो सदिशों को निरूपित करते हैं। निम्न में से कौन-सा ℝn पर आंतरिक गुणन (inner product) परिभाषित करता है?

More Mathematical Science Questions

Q1.The only function among the following that satisfies Cauchy's Riemann (C-R) equations is : 
Q2.What is the cardinality of the set of real solutions of ex + x = 1?
Q3.Let A be a 10 x 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true? 
Q4.Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 
Q5.Consider the contour γ given by \(\rm \gamma (\theta)=\left\{\begin{matrix}e^{2i \theta}&for\ \theta \in [0, \pi /2]\\\ 1+2e^{2i\theta}&for\ \theta \in [\pi/2, 3\pi/2]\\\ e^{2i\theta}&for\ \theta \in [3\pi/2, 2\pi]\end{matrix}\right.\) Then what is the value of \(\rm \int_{\gamma}\frac{dz}{z(z-2)}\)?
Q6.Let X1....X10 be a random sample from a distribution with the probability density function \(\rm f(x|θ)=\left\{\begin{matrix}θ x^{θ-1}, & if \:0<x<1\\\ 0, & otherwise,\end{matrix}\right.\) where θ > 0 Is an unknown parameter. The prior distribution of θ is given by   \(\rm \pi (θ)=\left\{\begin{matrix}θ e^{-θ}, & if\ θ>0\\\ 0, & otherwise,\end{matrix}\right.\) The Bayes estimator of θ under squared error loss is  
Q7.For a quadratic form f(x, y, z) ∈ ℝ[x, y z], we say that (a, b, c) ∈ ℝ3 is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)? 
Q8.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q9.Let X1, X2 be a random sample from a population having probability density function f ∈ { f0, f1} where  \(\rm f_0(x)=\left\{\begin{matrix}\frac{1}{2}& if\ 0\le x\le 2\\\ 0&otherwise\end{matrix}\right.\ and \ \rm f_1(x)=\left\{\begin{matrix}\frac{1}{4}& if\ 0\le x\le 4\\\ 0&otherwise\end{matrix}\right.\) For testing the null hypothesis H0 : f = f0 against the alternate hypothesis H1 : f = f1, the power of a most powerful test of size α = 0.05 is equal to  
Q10.Let V be the real vector space of 2 x 2 matrices with entries in ℝ. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where \(\rm A=\begin{pmatrix}2&0\\\ 0&1\end{pmatrix}\). What is the characteristic polynomial of T? 
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
support@testbook.com
Toll Free:1800 833 0800
Office Hours: 10 AM to 7 PM (all 7 days)
Company
About usCareers We are hiringTeach Online on TestbookPartnersMediaSitemap
Products
Test SeriesTestbook PassOnline CoursesOnline VideosPracticeBlogRefer & EarnBooks
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Hot Links： teen patti gold apk download master teen patti teen patti bliss teen patti master download teen patti game