What is the Complete Set of 2x2 Matrices Satisfying $A^2-4A+3I=0$?

  • MHB
  • Thread starter juantheron
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In summary: OK, I know it is bad form to reply to ones own posts, but:Let us restrict our attention to 2x2 matrices. We know that \(A\) satisfies its' charateristic equation, which we may write as: \(\lambda^2+b\lambda+c=0\) and so \(A^2+bA+cI=0\), now suppose it also satisfies another second degree equation: \(A^2+\beta A+\gamma I =0\) (which forces \(\beta \ne b\) if these are to be different and consistent).Then we have: \((b-\beta)A+(c-\gamma)I=0
  • #1
juantheron
247
1
Find all square matricx $A$ that satisfy the equation $A^2-4A+3I=0$

where $I$ is a Identity matrix
 
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  • #2
jacks said:
Find all square matricx $A$ that satisfy the equation $A^2-4A+3I=0$

where $I$ is a Identity matrix

Is...

$\displaystyle A^{2}- 4 A +3 I= A\ (A -4\ I + 3\ A^{-1})= A\ (A -3\ I - I + 3\ A^{-1})= A\ \{(A-3\ I)- A^{-1}\ (A-3\ I)\}= A\ (A-3\ I)\ (I-A^{-1})$

... so that the A satisfying the given equation also satisfy one of the equations...

$\displaystyle A-3\ I=0$

$\displaystyle A-I=0$

Kind regards

$\chi$ $\sigma$
 
  • #3
chisigma said:
Is...

$\displaystyle A^{2}- 4 A +3 I= A\ (A -4\ I + 3\ A^{-1})= A\ (A -3\ I - I + 3\ A^{-1})= A\ \{(A-3\ I)- A^{-1}\ (A-3\ I)\}= A\ (A-3\ I)\ (I-A^{-1})$

... so that the A satisfying the given equation also satisfy one of the equations...

$\displaystyle A-3\ I=0$

$\displaystyle A-I=0$

Kind regards

$\chi$ $\sigma$

The quadratic in A can be factorised straight off without having to worry about the existence of inverses. But I'm not sure that we can conclude that if a product is zero then one of the terms in the product is zero in this case.

CB
 
  • #4
CaptainBlack said:
The quadratic in A can be factorised straight off without having to worry about the existence of inverses. But I'm not sure that we can conclude that if a product is zero then one of the terms in the product is zero in this case.

CB

A trivial test confirms that $A=I$ and $A=3\ I$ are solutions of the equation...

$\displaystyle A^{2}-4\ A+3\ I=0$ (1)

The question is: there are some other solutions to (1)?... it is clear that the question is strictly connected to another question: can exist two matrices X and Y, with $X\ne 0$ and $Y\ne 0$, for whose is $X\ Y=0$?... a very interesting question!...

Kind regards

$\chi$ $\sigma$
 
  • #5
chisigma said:
A trivial test confirms that $A=I$ and $A=3\ I$ are solutions of the equation...

$\displaystyle A^{2}-4\ A+3\ I=0$ (1)

The question is: there are some other solutions to (1)?... it is clear that the question is strictly connected to another question: can exist two matrices X and Y, with $X\ne 0$ and $Y\ne 0$, for whose is $X\ Y=0$?... a very interesting question!...

Kind regards

$\chi$ $\sigma$

The answer to the last question is yes:
\[ A=\left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right] \]
Then \( A^2=0 \)

CB
 
  • #6
Thanks to all experts

but my doubt is , is there is any other matrix exists for which $A=\begin{bmatrix}a &b \\
c & d
\end{bmatrix}$ which satisfy $A^2-4A+3I=0$
 
  • #7
jacks said:
Thanks to all experts

but my doubt is , is there is any other matrix exists for which $A=\begin{bmatrix}a &b \\
c & d
\end{bmatrix}$ which satisfy $A^2-4A+3I=0$

Yes.
$$
A = \begin{bmatrix}3&0\\0&1\end{bmatrix}
$$

I just used the Cayley-Hamilton Theorem in reverse.
 
  • #8
dwsmith said:
Yes.
$$
A = \begin{bmatrix}3&0\\0&1\end{bmatrix}
$$

I just used the Cayley-Hamilton Theorem in reverse.

Which is that a matrix \(A\) satisfies its' own characteristics equation:

\[\det (A-\lambda I)=0\]

Which off course allows you to generate many 2x2 solutions of the original equation.

But is that (with the two we already have, which are not restricted to being 2x2) all of them?

CB
 
  • #9
CaptainBlack said:
Which is that a matrix \(A\) satisfies its' own characteristics equation:

\[\det (A-\lambda I)=0\]

Which off course allows you to generate many 2x2 solutions of the original equation.

But is that (with the two we already have, which are not restricted to being 2x2) all of them?

CB

OK, I know it is bad form to reply to ones own posts, but:

Let us restrict our attention to 2x2 matrices. We know that \(A\) satisfies its' charateristic equation, which we may write as: \(\lambda^2+b\lambda+c=0\) and so \(A^2+bA+cI=0\), now suppose it also satisfies another second degree equation: \(A^2+\beta A+\gamma I =0\) (which forces \(\beta \ne b\) if these are to be different and consistent).

Then we have: \((b-\beta)A+(c-\gamma)I=0\), and so: \(A=\frac{\gamma-c}{b-\beta}I\)

So the 2x2 solutions to \(A^2-4A+3I=0 \) are either matrices with charateristic equation \(\lambda^2-4\lambda+3=0\) or are multiples of the identity (and so of neccessity the solutions we found by factorising the equation)

CB
 
Last edited:

FAQ: What is the Complete Set of 2x2 Matrices Satisfying $A^2-4A+3I=0$?

What is the purpose of "Finding $A$ for $A^2-4A+3I=0$"?

The purpose of finding $A$ for $A^2-4A+3I=0$ is to solve for the matrix $A$ that satisfies the given equation. This will allow us to find the eigenvalues and eigenvectors of $A$, which are important concepts in linear algebra.

What is the relationship between $A$ and $A^2-4A+3I=0$?

$A$ is the unknown matrix that we are trying to find, while $A^2-4A+3I=0$ is an equation that involves the matrix $A$. The solution to this equation will give us the values of $A$.

How do we solve for $A$ in $A^2-4A+3I=0$?

To solve for $A$ in $A^2-4A+3I=0$, we can use various methods such as diagonalization, eigenvalue decomposition, or the Cayley-Hamilton theorem. These methods involve manipulating the given equation and solving for the values of $A$.

What are the possible values of $A$ in $A^2-4A+3I=0$?

There are infinitely many possible values of $A$ that satisfy $A^2-4A+3I=0$. These values depend on the method used to solve for $A$. Some methods may give real values, while others may give complex values.

Why is solving for $A$ in $A^2-4A+3I=0$ important?

Solving for $A$ in $A^2-4A+3I=0$ is important because it allows us to find the eigenvalues and eigenvectors of $A$, which have various applications in mathematics, physics, and engineering. It also helps us understand the properties of matrices and their behavior in different operations.

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