Make functions with custom autodiff `var<mat>` friendly

Description

This issue is part of https://github.com/stan-dev/math/issues/1805.

The goal is to convert every function with custom reverse mode autodiff to accept var_value<Eigen::Matrix<T, R, C>> types as arguments. The implementations should probably be based off https://github.com/stan-dev/math/issues/2018 (it's fine if they beat that in, I can just remove them from #2018).

Edit this list and check off functions as they're implemented in develop:

• softmax
• sum
• multiply
• elt_multiply #2121
• dot_self #2106
• columns_dot_self #2106
• determinant #2106
• inverse #2106
• log_determinant #2106
• multiply_lower_tri_self_transpose #2106
• tcrossprod #2106
• matrix_power #2106
• ordered_constrain #2106
• positive_ordered_constrain #2106
• simplex_constrain #2106
• unit_vector_constrain #2106
• var_value.transpose #2163
• dot_product #2106 and #2150
• variance #2168
• divide #2171
• elt_divide #2171
• operator+ #2115
• quad_form #2164
• trace_quad_form #2164
• columns_dot_product #2199
• rows_dot_prod #2199
• log_sum_exp #2106 (partial), #2169 (the rest)
• sd #2169
• log_softmax #2169
• operator- #2196
• squared_distance #2212
• ordered_constrain
• positive_ordered_constrain
• unit_vector_constrain
• log_determinant_spd #2232
• mdivide_left_spd #2213
• mdivide_left_tri #2213
• mdivide_left #2230
• exp #2254
• tanh #2254
• log1m #2254
• square #2254
• cholesky_corr_constrain #2254
• cholesky_factor_constrain #2254
• corr_matrix_constrain #2254
• cov_matrix_constrain_lkj #2254
• cov_matrix_constrain #2254
• read_cov_matrix #2254
• read_corr_matrix #2254
• read_corr_L #2254
• read_cov_L #2254
• log #2254
• inv_logit #2254
• multiply_log #2254
• dot_product #2305

In Progress:

• mdivide_left_ldlt #2266
• log_determinant_ldlt #2266
• trace_gen_inv_quad_form_ldlt #2266
• trace_inv_quad_form_ldlt #2266
• trace_gen_quad_form #2280
• trace #2284
• check_symmetric #2284
• matrix_normal_prec_log #2284
• matrix_normal_prec_lpdf #2284
• multi_gp_lpdf #2284
• multi_normal_lpdf #2284
• multi_normal_prec_lpdf #2284
• multi_student_t_lpdf #2284
• inv_wishart_lpdf #2284
• wishart_lpdf #2284
• simplex_constrain #2285
• lb_constrain #2285
• ub_constrain #2285
• lub_constrain #2285

Originally on the todo list but I'm taking them off for now:

• fma
• cov_exp_quad
• gp_exp_quad_cov
• gp_periodic_cov

Current Version:

v3.3.0

Start with the remaining function list, especially mdivide_left_ldlt, trace, check_symmetric, and the distribution functions, and review the related issues #2266, #2280, #2284, and #2285. Check how custom reverse-mode autodiff functions accept var_value<Eigen::Matrix<T, R, C>> arguments. Done means all intended functions support these arguments and the checklist is complete.