Cosine of matrix and matrix of cosines

Question

Suppose I have cosine of the matrix $$ \tag 1 \cos\left( \begin{pmatrix} a & b \\ c & d\end{pmatrix}\right) $$ May I write it in a form $$ \tag 2 \begin{pmatrix} \cos(a) & \cos(b) \\ \cos(c) & \cos(d)\end{pmatrix}? $$

Answer 1

The standard definition of cosine of a matrix, sometimes useful in thinking about differential equations, is $$ \cos A = I - \frac 1 2 A^2 + \frac 1 {24} A^4 - \frac 1 {720} A^6 + \cdots = \sum_{n=0}^\infty \frac {(-1)^n} {(2n)!} A^{2n}. $$ This is the same series that is the power expansion of $\cos A$ when $A$ is a real or complex number.