Matrix question help.

Question

Consider

$$X = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}.$$

Find a real matrix $A$ for which $A^2 = X$. I don't know how to answer this or where to start.

Answer 1

Write

$$A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}.$$

Then

$$A^2=\begin{bmatrix} a^2+bc & \cdot \\ \cdot & \cdot \end{bmatrix}.$$

I'll let you work out the expressions that should go into the dots inside the matrix. Then solve for $a,b,c,d$.

Answer 2

A surprising result is that the square root of a 2 x 2 matrix can be constructed, it is not unique, and since it involves two very relevant functions of a matrix I think it's worth presenting : $$ M=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

Let $ a + d = tr(M) $ , $ ad - bc = Det M $

$ s^2 = Det M $ , $ t^2 = tr(M) + 2s $

Then $ R = \frac{1}{t}\begin{bmatrix} a+s & b \\ c & d+s \end{bmatrix} $ is a square root of the matrix M.

See: http://en.wikipedia.org/wiki/Square_root_of_a_2_by_2_matrix