# 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.