I found this problem with matrices larger than 3x3.

Consider a matrix like this:
1 0 0 0
0 1 0 0
0 0 0 1
0 0 a 0

Its determinant is -a, but Calc says its 0.

Steps to reproduce:

1) emacs -Q
2) Start Calc: C-x * c
3) Enter algebraic mode by typing '
4) Enter the above matrix, like this:
[[1 0 0 0][0 1 0 0][0 0 0 1][0 0 a 0]]
5) Calculate the determinant: V D

Calc says that the determinant is 0, which is wrong.

If, however, a is replaced with a value, say 5, Calc correctly solves it
and reports the determinant as -5.

With a 3x3 matrix, the result is correct.  That is:
V D on [[1 0 0][0 0 1][0 a 0]] gives -a as the result.

I think the algorithm for choosing the pivots should be improved to be
able to permutate the rows and choose a symbolic variable as the pivot.

In GNU Emacs 28.0.50 (build 7, x86_64-pc-linux-gnu, GTK+ Version 3.22.30,
cairo version 1.15.10)
 of 2020-05-12 built on tbb-desktop
Repository revision: 703115829b35de6a90d7bafb7931f905e79d0d35
Repository branch: master
Windowing system distributor 'The X.Org Foundation', version 11.0.12005000
System Description: Ubuntu 18.04.4 LTS

Recent messages:
For information about GNU Emacs and the GNU system, type C-h C-a.
C-x *  (Type ? for a list of Calc options)
Welcome to the GNU Emacs Calculator!  Press ‘?’ or ‘h’ for help, ‘q’ to quit
Working... LUD step = 3/4
C-x *  (Type ? for a list of Calc options)

Configured features:
XPM JPEG TIFF GIF PNG CAIRO SOUND DBUS GSETTINGS GLIB NOTIFY INOTIFY
GNUTLS FREETYPE HARFBUZZ ZLIB TOOLKIT_SCROLL_BARS GTK3 X11 XDBE XIM
MODULES THREADS PDUMPER GMP

Important settings:
  value of $LC_MONETARY: es_AR.UTF-8
  value of $LC_NUMERIC: es_AR.UTF-8
  value of $LC_TIME: es_AR.UTF-8
  value of $LANG: en_US.UTF-8
  locale-coding-system: utf-8-unix

Major mode: Lisp Interaction

Minor modes in effect:
  tooltip-mode: t
  global-eldoc-mode: t
  eldoc-mode: t
  electric-indent-mode: t
  mouse-wheel-mode: t
  tool-bar-mode: t
  menu-bar-mode: t
  file-name-shadow-mode: t
  global-font-lock-mode: t
  font-lock-mode: t
  blink-cursor-mode: t
  auto-composition-mode: t
  auto-encryption-mode: t
  auto-compression-mode: t
  line-number-mode: t
  transient-mark-mode: t

Load-path shadows:
None found.

Features:
(shadow sort mail-extr emacsbug message rmc puny dired dired-loaddefs
format-spec rfc822 mml mml-sec password-cache epa derived epg epg-config
gnus-util rmail rmail-loaddefs text-property-search time-date subr-x seq
byte-opt gv bytecomp byte-compile cconv mm-decode mm-bodies mm-encode
mail-parse rfc2231 mailabbrev gmm-utils mailheader sendmail rfc2047
rfc2045 ietf-drums mm-util mail-prsvr mail-utils calc-arith calc-mtx
calccomp calc-alg calc-vec calc-aent calc-menu easymenu calc-misc
calc-prog calc-ext cl-loaddefs cl-lib calc calc-loaddefs calc-macs
tooltip eldoc electric uniquify ediff-hook vc-hooks lisp-float-type
mwheel term/x-win x-win term/common-win x-dnd tool-bar dnd fontset image
regexp-opt fringe tabulated-list replace newcomment text-mode elisp-mode
lisp-mode prog-mode register page tab-bar menu-bar rfn-eshadow isearch
timer select scroll-bar mouse jit-lock font-lock syntax facemenu
font-core term/tty-colors frame minibuffer cl-generic cham georgian
utf-8-lang misc-lang vietnamese tibetan thai tai-viet lao korean
japanese eucjp-ms cp51932 hebrew greek romanian slovak czech european
ethiopic indian cyrillic chinese composite charscript charprop
case-table epa-hook jka-cmpr-hook help simple abbrev obarray
cl-preloaded nadvice loaddefs button faces cus-face macroexp files
text-properties overlay sha1 md5 base64 format env code-pages mule
custom widget hashtable-print-readable backquote threads dbusbind
inotify dynamic-setting system-font-setting font-render-setting cairo
move-toolbar gtk x-toolkit x multi-tty make-network-process emacs)

Memory information:
((conses 16 76586 9357)
 (symbols 48 9259 1)
 (strings 32 22656 1028)
 (string-bytes 1 685264)
 (vectors 16 11989)
 (vector-slots 8 156067 9126)
 (floats 8 28 37)
 (intervals 56 207 0)
 (buffers 992 12))