The document contains three parts of a matrix-related problem. Let's break it down:

Part A:

1.

a. Define a skew-symmetric matrix with an example.

• A skew-symmetric matrix is a square matrix $A$ such that $A^T = -A$, meaning the transpose of the matrix is equal to the negative of the matrix itself.
• Example: 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{pmatrix}$$ Here, $$A^T = -A$$, as the transpose of $$A$$ will match the negative of $$A$$.

b. Show that $D = (a + b + c)^3$.

• In this question, matrix $D$ is given as: a-b-c & a & a \\ b-c-a & b & b \\ c-a-b & c & c \end{pmatrix}$$ You need to show that $$D = (a + b + c)^3$$. This requires expanding and simplifying the expression to prove the equality.

c. Find a matrix $B$ for matrix $A$ such that $AB = BA = I$.

• In this part, matrix $A$ is given as: 2 & 3 & 1 \\ 1 & 0 & -2 \\ 1 & -4 & 1 \end{pmatrix}$$ Here, you need to find matrix $$B$$ that satisfies the condition $$AB = BA = I$$, which means that $$B$$ is the inverse of $$A$$. To find $$B$$, you can compute the inverse of matrix $$A$$.

Would you like me to walk you through the solutions to these problems in detail? Or do you have any specific questions?