Testbook Logo
Exams SuperCoaching Test Series Skill Academy
More
English
Hindi
Telugu
Bengali

Question

Download Solution PDF

If A and B are two matrices such that AB = B and BA = A, then A2 + B2 is equal to

This question was previously asked in
UP TGT Mathematics 2021 Official Paper
Download PDF Attempt Online
View all UP TGT Papers >
  1. 2AB
  2. 2BA
  3. A + B
  4. AB

Answer (Detailed Solution Below)

Option 3 : A + B
Free Tests
View all Free tests >
Free
UP TGT Hindi FT 1
9.8 K Users
125 Questions 500 Marks 120 Mins
Start Now

Detailed Solution

Download Solution PDF

Concept:

The associative property of matrix is given by:

X (YZ) = (XY) Z      ----(1)

Given:

AB = B and BA = A      ----(2)

Calculation:

A2 + B2

⇒ AA + BB

⇒ A (BA) + B (AB)      [using (2)]

⇒ (AB) A + (BA) B      [using (1)]

⇒ BA + AB

⇒ A + B

Hence, A2 + B2 = A + B.

Download Solution PDF
Latest UP TGT Updates

Last updated on May 6, 2025

-> The UP TGT Exam for Advt. No. 01/2022 will be held on 21st & 22nd July 2025.

-> The UP TGT Notification (2022) was released for 3539 vacancies.

-> The UP TGT 2025 Notification is expected to be released soon. Over 38000 vacancies are expected to be announced for the recruitment of Teachers in Uttar Pradesh. 

-> Prepare for the exam using UP TGT Previous Year Papers.

More Operations on Matrices Questions

Q1.Find the additive inverse of matrix A = \(\left[\begin{array}{cc}2 & 1 \\ −3 & 0\end{array}\right]\)
Q2.The number of all possible matrices of order 2 × 2 with each entry 0 or 1 is:
Q3.On using elementary row operation R1 → R1 – 3R2 in the following matrix equation: \(\left[\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right] = \left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]\), we have :
Q4.On using elementary column operations C2 → C2 – 2C1 in the following matrix equation \(\left[\begin{array}{cc}1 & −3 \\ 2 & 4\end{array}\right] = \left[\begin{array}{cc}1 & −1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]\), we have :
Q5.For any two matrices A and B, we have
Q6.If A = \(\left[\begin{array}{ccc}2 & −1 & 3 \\ −4 & 5 & 1\end{array}\right]\) and B = \(\left[\begin{array}{cc}2 & 3 \\ 4 & −2 \\ 1 & 5\end{array}\right]\), then
Q7.If A and B are square matrices of the same order, then (A + B) (A – B) is equal to
Q8.If A = \(\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & \sin \theta & −\cos \theta\end{array}\right)\), then which of the following are correct? 1. A + adj A is a null matrix 2. A−1 + adj A is a null matrix 3. A − A−1 is a null matrix Select the correct answer using the code given below :
Q9.Let A and B be non-singular matrices of the same order such that AB = A and BA = B. Which of the following statements is/are correct ? 1. A2 = A 2. AB2 = A2B Select the correct answer using the code given below:
Q10.Consider the following statements in respect of square matrices A and B of same order : 1. If AB is a null matrix, then at least one of A and B is a null matrix. 2. If AB is an identity matrix, then BA = AB. Which of the above statements is/are correct?

More Matrices Questions

Q1.Let α be a root of the equation x2 + x+ 1= 0 and the matrix A = \(\frac{1}{\sqrt{3}}\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha^4 \end{array}\right]\) then the matrix A31 is equal to
Q2.Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 − B2A2)X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has:
Q3.Let α ∈ (0, α) and \(A = \begin{bmatrix} 1 & 2 & \alpha \\\ 1 & 0 & 1 \\\ 0 & 1 & 2 \end{bmatrix}\). If det(adj(2A - AT). adj(A - 2AT)) = 28, then (det(A))2 is equal to: 
Q4.For a 3 × 3 matrix A = [aij], whose element aij is given by aij = \(\rm \sqrt{i^2+3j}\) then what is element a31?
Q5.Let \(A=\left(\begin{array}{cc} α & 1 \\ 0 & -1 \end{array}\right)\) and \(B=\left(\begin{array}{ll} 4 & 1 \\ 0 & 1 \end{array}\right)\), such that A2 = B, then the value of α is:
Q6.The number of solutions of the matrix equation \(A^2=\left[\begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right]\) is:
Q7.If \(A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]\) and A + A' = I, then the value of α is
Q8.If the matrix \(\left[\begin{array}{lll}5 & 8 & 6 \\ 3 & 2 & 4 \\ 1 & 7 & 9\end{array}\right]\) is expressed in the form A + B, where A is symmetric and B is skew-symmetric, then the value of B is
Q9.Find the additive inverse of matrix A = \(\left[\begin{array}{cc}2 & 1 \\ −3 & 0\end{array}\right]\)
Q10.Let l ≥ 1 be a positive integer. What is the dimension of the ℝ-vector space of all polynomials in two variables over R having a total degree of at most l?
Crack Super Subscription Model with
India's Super Teachers
Explore Supercoaching For FREE
Suggested Test Series
View All >
UP TGT Mock Test Series 2025
444 Total Tests with 6 Free Tests
Start Free Test
Social Science (SST) Content for All Teaching Exams - Let's Crack TET!
191 Total Tests with   1 Free Tests
Start Free Test
Suggested Exams
Exam-Logo
NDA

NDA Important Links

More Mathematics Questions

Q1.log10 10000 = _______?
Q2.Consider a system equation \(2x+y-z=0\), \(4x - py + 4z = 4\) and \(x-y+z=q\) where \(p, q \in I\) and \(p, q \in [1, 10]\), then identify the correct statement(s).   List-I List-II (I) Number of ordered pairs \((p, q)\) for which system of equation has unique solution is (P) 1 (II) Number of ordered pairs \((p, q)\) for which system of equation has no solution is (Q) 9 (III) Number of ordered pairs \((p, q)\) for which system of equation has infinite solution is (R) 91 (IV) Number of ordered pairs \((p, q)\) for which system of equation has atleast one solution is (S) 90
Q3.Match List I with List II:   List - I List - II (I) The number of solutions of the equation\((1 - \cos 2x)(\cos 2x + \cot^2 x) = 0\) where \(0 \le x \le 2\pi\) is (P) 4 (II) \(\tan 81^\circ - \tan 63^\circ - \tan 27^\circ + \tan 9^\circ\) equals (Q) 3 (III) The number of solutions of (x, y), which satisfy |y| = sin x and y = cos-1(cos x), where -2π ≤ x ≤ 2π is (R) 2 (IV) For the equation \(\cos^{-1}x + \cos^{-1}2x + \pi = 0\), the number of real solutions is (S) 1     (T) 0  
Q4.If Σn = 36, find n
Q5.If log 2 = 0.2614, log 3 = 0.3521, log 6 = ?
Q6.The growth model for a population is Nt = N0ert where Nt is the population at time t, N0 is the initial population and r is the per capita growth rate. How long does it take for the population numbers to double?
Q7.Suppose that the growth rate of the number of individuals (N) in a population in an environment with a carrying capacity K is modelled as \(\frac{d N}{d t}=r N \frac{K-N}{K}\) where r is the per capita growth rate and t is time; r and K are positive constants. At which of the following values of N will the population growth rate be the maximum?
Q8.The shadow of a vertical pole, formed due to a nearby lamp which is at a height of 12 m, measures 4 m. If the lamp is moved horizontally 2 m further away from the pole, the length of the shadow measures 8 m. What is the height (in m) of the pole?
Q9.A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Out of these 16 number of cards, two cards are drawn at random. Find the probability that at least one of them is an ace :
Q10.If the equations 2x2 + kx - 5 = 0 and x2 - 3x - 4 = 0 have one root common, then find the value of k.
Important Exams
SuperCoaching
Exams
Test Series
Previous Year Papers
Objective Questions
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
[email protected]
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
Our Apps
Follow us on
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Get Free Access Now
Hot Links： teen patti 3a teen patti master downloadable content teen patti comfun card online teen patti bliss online teen patti real money