I have a bit of a problem with some matrix equations:
Task 1: $X,A$ arbitrary matrices, $A$ invertible. Solve for $X$: $XA+A^T=I \Longleftrightarrow XA = I-A^T \Longleftrightarrow X = IA^{-1}-A^TA^{-1} \Longleftrightarrow X = A^{-1}-A^TA^{-1}$
Task 2: Let $X,A,B$ be arbitrary matrices. Solve for $X$: $X^TC=[(2A(X+B)]^T \Longleftrightarrow X^TC=(2AX+2AB)^T \Longleftrightarrow X^TC=2A^TX^T+2A^T+B^T$
Task 3: $x\in \mathbb R^n,y \in \mathbb R^d, A \in \mathbb R^{d \times n}$, and $A^TA$ invertible. Solve for $X$: $(Ax-y)^TA=0 \Longleftrightarrow ((Ax)^T-(y)^T)A = 0 \Longleftrightarrow (Ax)^TA-(y)^TA=0 \Longleftrightarrow (A^T)(x^T)A-(y)^TA=0 \Longleftrightarrow A^TAx^TA-y^TA = 0 \Longleftrightarrow A^TAx^TA=y^TA \Longleftrightarrow x^TA=(A^TA)^{-1}y^TA$
Task 4: As above and $B\in \mathbb R^{n \times n}$,$A^TA$ invertible. Solve for $X$: $(Ax-y)^TA+x^TB=0 \Longleftrightarrow ((Ax)^T-(y)^T)A+x^TB = 0 \Longleftrightarrow A^Tx^TA-y^Ta+x^TB = 0$
So I tried solving these below each task and I'm quite sure Task 1 is correct but I always get stuck.
The error you are making is that $$(AB)^T\ne A^TB^T$$ Instead $$(AB)^T= B^TA^T$$ This will allow you to immediately solve tasks 2-4.