question about a diagonal matrix above $\mathbb{C}$

Question

• what is the number of different diagonal matrices from the series $2 \times 2$ over $\mathbb{C}$ that fulfill $D^3$= $-72I_2$?

• what is the modulus $r$ of an element in the diagonal of this matrix?

  I know that D should be a matrix that contains complex elements, and I know how to find the roots for $\sqrt[3]{-72}$ which are 3 roots (one of them is a Real number). so I guessed that there are two different diagonal matrices from the series $2\times 2$ above $\mathbb{C}$ that fulfill the conditions , and I know its wrong, so my second guess is $4$. secondly, calculating the modulus, which is $\sqrt{a^2+b^2}$ for the complex element in the matrix was wrong too. can someone do the calculations so I can know where I was wrong? or write the matrix D with its whole elements?

Answer 1

As you have established, the equation $x^3 + 72 = 0$ as three solutions. I call these solutions $x_1,x_2,x_3$. We note that for any $2 \times 2$ diagonal matrix $$ D = \pmatrix{d_1&0\\0&d_2}, $$ we have $$ D^3 + 72I = \pmatrix{d_1^3 + 72 & 0\\0&d_2^3 + 72} $$ so, the matrix $D$ will satisfy $D^3 + 72I = 0$ if and only if $d_1$ and $d_2$ are both solutions to the equation $x^3 + 72 = 0$.

Since there are $3$ possible values of $d_1$ and $3$ possible values of $d_2$ that may be chosen independently, the total number of such matrices $D$ is $3 \times 3 = 9$.

That is, the valid choices of $D$ are $$ \pmatrix{x_1 & 0\\0&x_1}, \pmatrix{x_1 & 0\\0&x_2}, \pmatrix{x_1 & 0\\0&x_3},\\ \pmatrix{x_2 & 0\\0&x_1}, \pmatrix{x_2 & 0\\0&x_2}, \pmatrix{x_2 & 0\\0&x_3},\\ \pmatrix{x_3 & 0\\0&x_1}, \pmatrix{x_3 & 0\\0&x_2}, \pmatrix{x_3 & 0\\0&x_3} $$

Note that each of these $x_i$ satisfy $|x_i|^3 = 72 \implies |x_i| = \sqrt[3]{72}$.

Answer 2

As you said $D$ is diagonal and an element of its diagonal is a cubic root of $-72$ so to construct $D$ we should choose (with possible repetition) $2$ elements from the $3$ roots so we have $3^2$ matrices. The module $r$ of every roots is $r=\sqrt[3]{72}$.