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Give a quick reason why each of these statements is true: (a) Every positive definite

Chapter 6, Problem 6.2.23

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QUESTION:

Give a quick reason why each of these statements is true: (a) Every positive definite matrix is invertible. (b) The only positive definite projection matrix is P = I. (c) A diagonal matrix with positive diagonal entries is positive definite. (d) A symmetric matrix with a positive determinant might not be positive definite!

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QUESTION:

Give a quick reason why each of these statements is true: (a) Every positive definite matrix is invertible. (b) The only positive definite projection matrix is P = I. (c) A diagonal matrix with positive diagonal entries is positive definite. (d) A symmetric matrix with a positive determinant might not be positive definite!

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We have to give the reasons why the given statements are true.

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