91Ó°ÊÓ

Let \(A\) be a square matrix of order \(n .\) (a) Show that \(\frac{1}{2}\left(A+A^{T}\right)\) is symmetric. (b) Show that \(\frac{1}{2}\left(A-A^{T}\right)\) is skew-symmetric. (c) Prove that \(A\) can be written as the sum of a symmetric matrix \(B\) and a skew-symmetric matrix \(C\) \(A=B+C\). (d) Write the matrix below as the sum of a symmetric matrix and a skew- symmetric matrix. \(A=\left[\begin{array}{rrr}2 & 5 & 3 \\ -3 & 6 & 0 \\ 4 & 1 & 1\end{array}\right]\).

Explain in your own words how to write a system of three linear equations in three variables as a matrix equation, \(A \mathbf{x}=\mathbf{b},\) as well as how to solve the system using an inverse matrix.

The columns of matrix \(T\) show the coordinates of the vertices of a triangle. Matrix \(A\) is a transformation matrix. \(A=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right], \quad T=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 4 & 2\end{array}\right]\) (a) Find \(A T\) and \(A A T\). Then sketch the original triangle and the two transformed triangles. What transformation does \(A\) represent? (b) A triangle is determined by \(A A T\). Describe the transformation process that produces the triangle determined by \(A T\) and then the triangle determined by \(T\)