Linear Algebra – How to Solve Equations with Matrices

linear algebramatricesmatrix equations

can you please give me some hints to solve the following? I really don't know how to start.
$$X^2= \begin{pmatrix}
6 & 2 \\ 3 & 7
\end{pmatrix}.$$
I tried to express this matrix as $4\cdot I + \begin{pmatrix}
2 & 2 \\ 3 & 3
\end{pmatrix}$ And somehow solve it, but I really have no clue. Please some help.

Best Answer

One simple but laborious way of doing it is as follows

Let $$X = \begin{bmatrix}a & b \\c & d\end{bmatrix}$$

Now, using the Caley-Hamilton Theorem, we have

$$X^2 - (a+d)X + |X|I = 0$$

Now, from the equation, we have $|X| = \pm 6$

Hence, we will have two sets of solutions each corresponding to one of these

$$|X| = 6$$

$$\implies X = \frac{1}{a+d}\begin{bmatrix}12 & 2 \\3 & 13\end{bmatrix}$$

Using the equivalence of the two representations of $X$, we have

$$a = \frac{12}{5} \\ b = \frac{2}{5} \\ c = \frac{3}{5} \\ d = \frac{13}{5}$$

Another solution to this is when each of the entries is negative. Similarly complete it for $|X| = -6$ to get the full set of solutions

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