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A तथा B को 2 × 2 के आव्यूह मानें। तब निम्न में से कौन-से सत्य है?

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. det(A + B) + det(A - B) = det A + det B 
  2. det(A + B) + det(A - B) = 2det A - 2det B 
  3. det(A + B) + det(A - B) = 2det A + 2det B 
  4. det(A + B) - det(A - B) = 2det A - 2det B 

Answer (Detailed Solution Below)

Option 3 : det(A + B) + det(A - B) = 2det A + 2det B 
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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.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?
Q3.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?
Q4.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?
Q5.यदि \(A=\left(\begin{array}{ll}3 & -2 \\ 2 & -1\end{array}\right)\), तब A20 के बराबर 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.A को वास्तविक प्रविष्टियों वाली 3 × 3 आव्यूह मानें। निम्न में से कौन-सा वक्तव्य असत्य है?
Q10.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.What is the cardinality of the set of real solutions of ex + x = 1?
Q2.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? 
Q3.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 
Q4.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 \(\rm \int_{\gamma}\frac{dz}{z(z-2)}\)?
Q5.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  
Q6.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)? 
Q7.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q8.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  
Q9.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? 
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