GATE CSE 2016 | Set 1 | Question: 05

Question

Two eigenvalues of a $3 \times 3$ real matrix $P$ are $(2+\sqrt {-1})$ and $3$. The determinant of $P$ is _______

Comments

Imaginary values exists in pairs, itself and conjugate

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Important note is - It's given that the matrix is real. Real matrix will have a real determinant. And hence, complex roots of the characterstics equation will come in pairs

Answer 1

Eigen values are roots of Characteristic equation $|A - \lambda I | = 0.$

For a $3×3$ matrix, characteristic equation will be cubic, so will have $3$ roots. Two roots are given as: $ 2 + i$ and $3$ and We know that complex roots always occur in pairs so, if $2+i$ is a root of characteristic equation, then $2-i$ must be other root.

$\lambda_{1} = 2+i$, $\lambda_{2} = 2-i$ and $\lambda_{3} = 3$

$\color{blue}{\det(A) = \lambda_{1}\lambda_{2}\lambda_{3} = (2+i)*(2-i)*3 = (2^2 - i^2)*3 = 5*3 = 15}$

Comments

https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem

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again pooved that NAT has a hidden concept to catch.

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if one eigen value is complex then they always occur in pair.

so we get all the 3 eigen values

now ,

sum of eigen values = trace of matrix which  does not helps in finding determinant

while

product of eigen values = determinant

Answer 2

Given two eigen values are (2+i) and 3.. since it is a real matrix the 3rd eigen value is 2-i
determinant of P = product of eigen values.
Solving we get,

Answer 15.

Comments

dear friend

for ANY matrix,

PRODUCT OF EIGEN VALUES = DETERMINANT OF THAT MATRIX ITSELF

any way in diagonal/triangular matrices determinant is product of the elements in principle diagonal therefore they are same as product of eigen values.

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what is the meaning of real matrix here?

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means there is no imaginary value in the matrix.

Answer 3

The determinant of a real matrix can never be imaginary. So, if one eigen value is complex, the other eigen value has to be its conjugate.   So, the eigen values of the matrix will be 2+i, 2-i and 3.   Also, determinant is the product of all eigen values. So, the required answer is (2+i)*(2-i)*(3) = (4-i²)*(3) = (5)*(3) = 15.

Answer 4

We know that for any real matrix the determinant must be real.

Now the determinant is product of eigen values, and if one of the eigen value is imaginary then to make it real it can only be multiplied by its conjugate.

So, the other eigen value must be conjugate of the first imaginary eigen value.

Hence, we can easily find the third eigen value.

Answer 5

Then conjugate of (2+i) is 2-i and it is the third value of diagonal 

We know that Product of diagonal elements is Value of Determinant

(2+i )*3*(2-i) = (4 -(-1))*3 = 15

Answer 6

Complex number exists in pairs

 λ1 = 2+i, λ2=2−i and λ3=3

det(P) = λ1λ2λ3 = (2+i)(2-i)(3) = 15 {i^2 = -1}