Symmetry is an important concept in mathematics. In prior units of Core-Plus Mathematics, you examined geometric shapes and graphs of functions in terms of their symmetries. Symmetry also applies to matrices, but only to square matrices. A square matrix is said to be symmetric if it has reflection (or mirror) symmetry about its main diagonal. (Recall that the main diagonal of a square matrix is the diagonal line of entries running from the top-left to the bottom-right corner.) So, a square matrix is symmetric if the numbers in the mirror-image positions, reflected in the main diagonal, are the same. For example, consider the three matrices below. Matrices A and B are symmetric, but matrix C is not symmetric.

$A = \left[ \begin{array} { l l l l } { 0 } & { 1 } & { 0 } & { 1 } \\ { 1 } & { 0 } & { 1 } & { 1 } \\ { 0 } & { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 0 } & { 0 } \end{array} \right]$

,

$B = \left[ \begin{array} { c c c c } { 25 } & { 3 } & { 4 } & { 5 } \\ { 3 } & { 36 } & { 6 } & { 7 } \\ { 4 } & { 6 } & { 9 } & { 8 } \\ { 5 } & { 7 } & { 8 } & { 10 } \end{array} \right]$

,

$C = \left[ \begin{array} { l l l l } { 0 } & { 0 } & { 1 } & { 1 } \\ { 1 } & { 0 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 1 } & { 0 } \end{array} \right]$

. a. Identify two square matrices from this lesson. b. Which of the following matrices are symmetric? For those that are not symmetric, explain why not. i. The pottery matrix, ii. The degree-of-difference matrix, iii. The movie matrix, iv. The loan matrix, v. The nonshooting-performance-statistics matrix, vi. The mutual-friends matrix. c. For those matrices in Part b that are symmetric, what is it about the situations represented by the matrix that causes the matrix to be symmetric? d. Create your own symmetric matrix with four rows and four columns. i. Compare the first row to the first column. Compare the second row to the second column. Do the same for the remaining two rows and columns. ii. Make a conjecture about the corresponding rows and columns of a symmetric matrix. e. Test your conjecture from Part d on the symmetric matrices you identified in Part b.