Eigenvalues and eigenvectors, 3x3 matrix using Remainder and Factor Theorem

In summary, the conversation is discussing finding the values of λ in a 3x3 matrix using the remainder and factor theorem. The attempt at a solution involves factoring and finding the roots of a quadratic equation.
  • #1
Jowin86
32
0

Homework Statement



i = the 3x3 matrix below

2-λ 0 1
-1 4-λ -1
-1 2 0-λ

Using remainder and factor theorem find the 3 values of λ.

Homework Equations



|i| = a1|b2c3-c2b3|-a2|a2c3-c2a3|+a3|a2b3-b2a3|

|a|=ad-bc

The Attempt at a Solution

(2-λ) |(4-λ x 0-λ)-(-1x2)|+1|(-1x2)-(4-λ x -1) **because b1 is 0 I've left it out**

(2-λ)[(4-λ)(0-λ)+2] +1 [-2+(4-λ)]

(2-λ)(4-λ)(0-λ)+1(4-λ)

Factorise out (4-λ):

(4-λ)[(2-λ)(0-λ)+1]

Multiply [] brackets out (FOIL):

(4-λ)(λ2-2λ+1)

...and now I'm stuck which probably means I've gona wrong somewhere ;-(

*should I have taken out 0-λ?

*on the 2nd line of my attempted answer I figured the +2 in the first square brackets canceled out the -2 in the second square brackets, was I wrong?Thanks for any hints and help :)
 
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  • #2
2-2λ+1) can be factored.

If you can't do that, then find the roots: λ2-2λ+1 = 0
 
  • #3
(4-λ) λ(-2λ)

λ=4 λ=2 λ=0

Is that wrong?

I'm not sure how to find the roots of λ2-2λ+1=0 ?
 
  • #4
Jowin86 said:
(4-λ) λ(-2λ)

λ=4 λ=2 λ=0

Is that wrong?
Yes, it's wrong
I'm not sure how to find the roots of λ2-2λ+1=0 ?

It's a quadratic equation. Solve it.
 
  • #5


Hello,

I would like to provide some guidance on your attempt at solving the problem.

Firstly, it is important to understand the concepts of eigenvalues and eigenvectors before attempting to solve a problem involving them. Eigenvalues are scalar values that represent the scaling factor of the eigenvector when multiplied by a matrix. Eigenvectors are non-zero vectors that are unchanged in direction when multiplied by a matrix.

In this problem, you are given a 3x3 matrix and asked to find the three values of λ using the remainder and factor theorem. The remainder and factor theorem states that if a polynomial f(x) is divided by (x-a), the remainder will be f(a). In other words, if we divide a polynomial by one of its roots, the remainder will be 0.

Now, let's look at your attempt at solving the problem. You correctly identified the matrix and set up the equation using the determinant formula. However, you made a mistake in the second line where you multiplied the two square brackets. This is incorrect because the two terms inside the square brackets do not have a common factor. Instead, you should have expanded the brackets to get:

(2-λ)[(4-λ)(0-λ)+2] +1 [-2+(4-λ)]

= (2-λ)[4λ-4λ2+2] +1 [-2+4-λ]

= (2-λ)(-4λ2+4λ+2) +1 (-λ+2)

= (-8λ2+8λ+4)-(4λ3-4λ2+2λ)+(λ-2)

= 4λ3-12λ2+7λ+2

Now, we can use the remainder and factor theorem to find the values of λ. Since we have a polynomial of degree 3, we need to find three values of λ. We can do this by setting the polynomial equal to 0 and solving for λ.

4λ3-12λ2+7λ+2 = 0

We can use synthetic division to find the roots of the polynomial. After dividing by (λ-1), we get:

λ3-3λ2+4λ-2 = 0

Now, we can use the quadratic formula to find the other two roots. Solving for λ, we get:

λ = 1, (3±√5)/2

Therefore,
 

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used in linear algebra to describe the behavior of a linear transformation or a matrix. Eigenvalues are scalar values that represent the amount by which an eigenvector is scaled when the linear transformation is applied. Eigenvectors are non-zero vectors that are transformed only by a scalar multiple of their original value.

What is a 3x3 matrix?

A 3x3 matrix is a matrix with 3 rows and 3 columns. It can be represented as a square array of numbers or variables. Matrices are used to efficiently perform calculations and solve equations in linear algebra, physics, and other fields of science and engineering.

What is the Remainder Theorem?

The Remainder Theorem is a mathematical concept that states that when a polynomial function is divided by a linear function, the remainder is equal to the value of the polynomial function at the value of the variable in the linear function. In other words, the remainder is the value left over after the division process is complete.

What is the Factor Theorem?

The Factor Theorem is a mathematical concept that states that if a polynomial function has a root or zero at a particular value of the variable, then the polynomial can be factored into linear and quadratic factors with that value as one of the roots. In other words, the factor theorem helps to find the factors of a polynomial function.

How are eigenvalues and eigenvectors used in a 3x3 matrix using the Remainder and Factor Theorem?

In a 3x3 matrix, the eigenvalues and eigenvectors can be calculated using the Remainder and Factor Theorem. The eigenvalues can be found by solving the characteristic equation of the matrix, which is obtained by setting the determinant of the matrix minus a scalar value equal to zero. The eigenvectors can then be found by substituting the eigenvalues into the matrix and solving for the corresponding eigenvectors. These values can be used to simplify calculations and solve equations involving the matrix.

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