Find all real matrices such that $X^{3}-4X^{2}+5X=\begin{pmatrix} 10 & 20 \\ 5 & 10 \end{pmatrix}$

Question

The following question come from the 1998 Romanian Mathematical Competition:

Find all matrices in $M_2(\mathbb R)$ such that $$X^{3}-4X^{2}+5X=\begin{pmatrix} 10 & 20 \\ 5 & 10 \end{pmatrix}$$

Can you guy please help me? Thanks a lot!

Answer 1

There is a matrix $U$ such that $$U^{-1}\begin{pmatrix} 10 & 20 \\ 5 & 10 \end{pmatrix} U=\begin{pmatrix} 20 & 0 \\ 0&0 \end{pmatrix}.$$

Let $Y=U^{-1}XU$, then $Y^{3}-4Y^{2}+5Y=\begin{pmatrix} 20 & 0 \\0 & 0 \end{pmatrix}$. The matrix $Y$ then commutes with $\begin{pmatrix} 20 & 0 \\ 0&0 \end{pmatrix}$ and so is diagonal. Let $Y=\begin{pmatrix} a & 0 \\ 0&b \end{pmatrix}.$

Then $a^{3}-4a^{2}+5a=20,b^{3}-4b^{2}+5b=0.$ The only real solutions are $a=4,b=0.$

Then $X$ is uniquely determined as $UYU^{-1}$. If we did not know that $X=\begin{pmatrix} 2 & 4 \cr 1 & 2 \end{pmatrix}$ we could find it using the matrix $U$.

Answer 2

Let $X=\begin{pmatrix} a & b \cr c & d \end{pmatrix}$. Then the matrix equation immediately gives that $d=a$ and $b=4c$. Now it is easy to solve the last equations in $a$ and $c$. The only real solution is $a=2$ and $c=1$, and hence $$ X=\begin{pmatrix} 2 & 4 \cr 1 & 2 \end{pmatrix}. $$ For complex numbers there are several other solutions, e.g., $a=\frac{i + 6}{2},\; c= \frac{2- i }{4}$, or $a=-\frac{\sqrt{-5}}{2},\;c=-\frac{\sqrt{-5}}{4}$.

Answer 3

Let $p$ be the polynomial in question and $R$ be the right hand side. Note that $R$ is equivalent to $D=\operatorname{diag} (20,0)$.

Let $V^{-1}RV = D$, then since $V^{-1}p(X)V = p(V^{-1}XV) = D$, we can look for solutions to $p(X)=D$ and then conjugate back to get the original solutions.

Note that $De_1 = 20 e_1, D e_2 = 0$. Hence $p(X)e_1 = 20e_1$, $p(X)e_2 = 0$.

If $\lambda$ is an eigenvalue of $X$ then $p(\lambda)$ is an eigenvalue of $p(X)$, hence $X$ has distinct eigenvalues and $p(\lambda_1) = 20, p(\lambda_1) = 0$. Hence $e_1,e_2$ are eigenvectors of $X$ (this is the key here).

In particular, $X$ is diagonal, so the problem reduces to solving $p(x) = 0$ (roots $0, 2 \pm i$) to get $X_{22}$ and $p(x)=20$ (roots $4,\pm \sqrt{5}i$) to get $X_{11}$ and seeing what combinations work.

Since the matrix is real, we see that $X$ must have roots $4,0$ and so $X = \operatorname{diag} (4,0)$.

To finish, we need to conjugate, if we let $V= \begin{bmatrix} 2 & -2 \\ 1 & 1\end{bmatrix}$, then $V X V^{-1} = \begin{bmatrix} 2 & 4 \\ 1 & 2\end{bmatrix}$.

Answer 4

Here is an other possibility to proceed. The Romanian NMO "should not know" the linear algebra related to diagonalization and/or Jordan forms for matrices, but for $2\times 2$ matrices it is a standard idea to use Cayley-Hamilton, since having the trace $t$ and the determinant $d$ of a matrix $A$ it is an exercise for matrix operations (theoretically also done in the classes) to check $A^2-tA+d=0$. In this sense, we may work as follows, using as much as possibly the arithmetics of the polynomial ring $\Bbb R[x]$.

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The given matrix $A$ with entries $10, 20, 5, 10$ has trace $20$, and determinant zero. Let $g$ be the characteristic polynomial of $A$, so $g(A) = A^2-20A=0$. The unknown matrix $X$ satisfies for the polynomial $f(x)=x^3-4x^2+5x$ the given relation $f(x)=A$. So $$ \begin{aligned} h(x):=g(f(x)) &=x^2(x^2 - 4x + 5)^2 - 20x(x^2 - 4x + 5) \\ &=(x^2 - 4x + 5)(x^2 + 5)(x - 4)x \end{aligned} $$ annihilates $X$.

Let $p\in\Bbb R[x]$ be the (monic) minimal polynomial of $X$.

It has degree two, (else we get a contradiction with $f(X)=A$,) so it divides $h$.

• The first factor is excluded immediately as a value for $p$, because this would imply $A=f(X)=Xp(X)=0$.

• The second factor is also excluded as a value for $p$, because else $f(x)-20=(x^2+5)(x-4)$ has the factor $(x^2+5)$, so $A-20I=(f-20)(A) =0$, again a contradiction.

• It follows $p(x)=x(x-4)=x^2-4x$. The rest obtained by division with rest of $f(x)=\color{gray}{x^3-4x^2}+5x$ by $p(x)=x^2-4x$ is $5x$, so we obtain: $$ A=f(X)=5X\ . $$ This brings the only matrix operation in the game $$ X=\frac 15A=\begin{bmatrix}2&4\\1&2\end{bmatrix}\ . $$

Answer 5

dan_fulea mentioned in another answer that the contestants are not expected to know diagonalisation or Jordan form. So, I will give a more elementary solution below. Let $$ A=uv^T=\pmatrix{2\\ 1}\pmatrix{1&2}. $$ The equation in question is equivalent to $$ X^3-4X^2+5X=5A.\tag{1} $$ One can easily verify that $A^2=4A$ and $X=A$ is a solution to $(1)$. In general, if $X$ satisfies $(1)$, we must have $XA=AX$, i.e. $Xuv^T=uv^TX$. Therefore $Xu=ku$ and $v^TX=kv^T$ for some common real factor $k$, and $XA=AX=kA$. It follows from $(1)$ that \begin{aligned} X^3A-4X^2A+5XA&=5A^2,\\ k^3A-4k^2A+5kA&=20A,\\ k^3-4k^2+5k-20&=0,\\ (k-4)(k^2+5)&=0. \end{aligned} Therefore $k=4$ and $XA=AX=4A$. Since $A^2=4A$, if we put $Y=X-A$, we get $YA=AY=0$ or $Yuv^T=uv^TY=0$. Hence $Y$ must be a real scalar multiple of $$ B=\pmatrix{2\\ -1}\pmatrix{1&-2} $$ and $X=A+bB$ for some real scalar $b$. As $X=A$ is a solution to $(1)$, $AB=BA=0$ and $B^2=4B$, if we substitute $X=A+bB$ into $(1)$, we get \begin{aligned} b^3B^3-4b^2B^2+5bB&=0,\\ 16b^3-16b^2+5b&=0,\\ b(16^2-16b+5)&=0,\\ b&=0. \end{aligned} Hence the only solution to $(1)$ is given by $X=A$.