Let $𝐴$ be an $𝑛 × 𝑛$ real matrix. Consider the following statements.
$(I)$ If $𝐴$ is symmetric, then there exists $𝑐 ≥ 0$ such that $𝐴 + 𝑐𝐼_𝑛$ is symmetric and positive definite, where $𝐼_𝑛$ is the $𝑛 × 𝑛$ identity matrix
$(II)$ If $𝐴$ is symmetric and positive definite, then there exists a symmetric and positive definite matrix $𝐵$ such that $𝐴 = 𝐵^2$ .
Which of the above statements is/are true?
(A) Only (I)
(B) Only (II)
(C) Both (I) and (II)
(D) Neither (I) nor (II)
