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Zeroes of a Polynomial

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Question

Find the total number of possible square matrix A of the order 3 with all real entries, whose adjoint matrix B has characterstics polynomial equation as x³ - x² + x +1 = 0

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Solution

The given polynomial have only one real root other roots are complex.

so no of matrix = 1

only one matrix can be made with one element.

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Similar questions

A

be a square matrix of order

3

whose all entries are

1

I_{3}

be the identity matrix of order

3

. Then the matrix

A - 3 I_{3}

P

is a non null matrix of order

3

with real number as entries, such that

P^{3} = 0,

O

is null matrix of order

3

I

is the identity matrix of order

3

)

Q. What is the number of the possible square matrix of order

3

with each entry

0

1

? If your answer is

2^{n}

n

If all the real numbers

x_{1}, x_{2}, x_{3}

, satisfying the equation

x^{3} - x^{2} + β x + γ = 0

Then, all possible values of

γ

Let A be a square matrix of order 2 or 3 and I will be the identity matrix of the same order. Then the matrix A -

λ

I is called the characteristic matrix of the matrix A, where

λ

is some complex number. The determinant of the characteristic matrix is called characteristic determinant of matrix A which will, of course, be a polynomial of degree 3 in

λ

. The equation det (A -

λ

=

0 is called the characteristic equation of the matrix A and its roots (the values of

λ

) are called characteristic roots or eigenvalues. It is also known that every square matrix has its characteristic equation.

The eigenvalues of the matrix

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 1 & 1 2 & 3 & 4 - 1 & - 1 & - 2 \end{matrix} ⎤ ⎥ ⎦

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Zeroes of a Polynomial

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