Understand the problem

Let A be a 3 x 3 real matrix with all the diagoanl entries equal to 0 . If 1 + i is an eigen value of A , the determinant of A equal ?
Source of the problem
Sample Questions (MMA ) :2019
Topic
Linear Algebra
Difficulty Level
Medium
Suggested Book
Schaum’s Outline of Linear Algebra

Start with hints

Do you really need a hint? Try it first!

Let A be a 3 x 3 real matrix with trace 0 Now( 1+i) be an eigen value . (Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values, proper values, or latent roots & Eigen vectors are In linear algebra,) (An eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation )
So ,( 1 + i) be the roots of the characteristic poly of A Now A is a real matrix, char poly of A \(\epsilon \mathbb{R}[x]\) [Right!] Therefore ( 1 – i) is also a root of char poly of A
deg( char poly of A ) =3 So , it has two imaginary roots & one real root Let real root be r Note tr(A) = 0 => r + 1 +i +1 -i = 0 , => r = -2 Can you play with the determinants now ?
We know the determinant of A is the product of all eigen values (-2)(1+i)(1-i) = detA => det(A) = -2[1 +1] = -4

Watch the video

Connected Program at Cheenta

College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

Similar Problems

Application of eigenvalue in degree 3 polynomial: ISI MMA 2018 Question 14

This is a cute and interesting problem based on application of eigen values in 3 degree polynomial .Here we are finding the determinant value .

Order of rings: TIFR GS 2018 Part B Problem 12

This problem is a cute and simple application on the ring theory in the abstract algebra section. It appeared in TIFR GS 2018.

Last three digit of the last year: TIFR GS 2018 Part B Problem 9

This problem is a cute and simple application on the number theory in classical algebra portion. It appeared in TIFR GS 2018.

Group in graphs or graphs in groups ;): TIFR GS 2018 Part A Problem 24

This problem is a cute and simple application on the graphs in groups in the abstract algebra section. It appeared in TIFR GS 2018.

Problems on quadratic roots: ISI MMA 2018 Question 9

This problem is a cute and simple application on the problems on quadratic roots in classical algebra,. It appeared in TIFR GS 2018.

Are juniors countable if seniors are?: TIFR GS 2018 Part A Problem 21

This problem is a cute and simple application on the order of a countable groups in the abstract algebra section. It appeared in TIFR GS 2018.

Coloring problems: ISI MMA 2018 Question 10

This problem is a cute and simple application of the rule of product or multiplication principle in combinatorics,. It appeared in TIFR GS 2018.

Diagonilazibility in triangular matrix: TIFR GS 2018 Part A Problem 20

This problem is a cute and simple application on the diagonilazibility in triangular matrix in the abstract algebra section. It appeared in TIFR GS 2018.

Multiplicative group from fields: TIFR GS 2018 Part A Problem 17

This problem is a cute and simple application on the Multiplicative group from fields in the abstract algebra section. It appeared in TIFR GS 2018.

Group with Quotient : TIFR GS 2018 Part A Problem 16

This problem is a cute and simple application on Group theory in the abstract algebra section. It appeared in TIFR GS 2018.