Question

Find the nth root of the matrix
b. an nth root of a matrix b is a matrix a such that an =
b. b = 25 0 0 4 , n = 2

Answer 1

$\mathbf B=\begin{bmatrix}25&0\\0&4\end{bmatrix}$

Since $\mathbf B$ is diagonal, we have


$\mathbf B^k=\begin{bmatrix}25^k&0\\0&4^k\end{bmatrix}$

Take $k=\dfrac12$. Then

$\mathbf A=\mathbf B^{1/2}=\begin{bmatrix}5&0\\0&2\end{bmatrix}$
$\implies\mathbf A^2=\mathbf B$