Q69Section C (Part C)69 of 120

Surjectivity of the derivative of $X\mapsto XX^T$

Let $M_2(\mathbb R)$ denote the space of real $2\times2$ matrices. Let $S$ be the vector subspace of $M_2(\mathbb R)$ comprising of all symmetric matrices. Let $F:M_2(\mathbb R)\to S$ be the map defined by $F(X)=XX^T$ . Let $DF_A:M_2(\mathbb R)\to S$ be the derivative of $F$ at $A\in M_2(\mathbb R)$ . Which of the following statements are true?

Multiple select: one or more options are correct. Select all of them, then submit — marks need the exact set.

Select every correct option, then submit.

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