Trying to align gap between two equations on a middle line with the equals signs of singular equations on first and third lines

Your question isn't very clear as the code has 3 lines, but the target image has 4.

Assuming the image is what you want, possibly something like this:

% Source - https://tex.stackexchange.com/q/760764
% Posted by user436553
% Retrieved 2026-03-12, License - CC BY-SA 4.0

\documentclass[12pt]{article}
\usepackage[ignoreall,a4paper,margin=1cm]{geometry}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{newtx}
\usepackage{setspace}
\setstretch{1.3}
\pagestyle{empty}

\begin{document}
\abovedisplayskip=0pt
\belowdisplayskip=0pt
\abovedisplayshortskip=0pt
\belowdisplayshortskip=0pt
\large
%% Remember the & for alignment...
\begin{align*}
  \mathbf{a}-2\mathbf{b}+\mathbf{c}&=\mathbf{0} \\
  \begin{aligned}
    \mathbf{a} &= 4\mathbf{i} - \mathbf{j} + \mathbf{k} \\
    \mathbf{a}&=
    \begin{pmatrix*}[r]
      4 \\
      -1 \\
      1 \\
    \end{pmatrix*}
  \end{aligned}
  \qquad &\phantom{=} \qquad
  \begin{aligned}
    \mathbf{b} &= \mathbf{i} - 2\mathbf{k} \\
    \mathbf{b}&=
    \begin{pmatrix*}[r]
      1 \\
      0 \\
      -2 \\
    \end{pmatrix*}
  \end{aligned}\\
  \mathbf{a}-2\mathbf{b}+\mathbf{c}&=\mathbf{0}
\end{align*}
\end{document}

Off-topic notes:

• times is obsolete and ought not be used; ditto mathptmx

  • I substituted newtx, which should be loaded after amssymb, if the latter is also required.

• mathtools loads amsmath.

• Use setspace and \setstretch rather than \linespread.

• Generally, bold should be used in maths when it carries semantic meaning. Generally, it is better in that case to create an appropriately named macro for yourself, rather than writing \mathbf everywhere. Then it is easy to change if you end up needing something different.

  • \NewDocumentCommand \something {m} {\mathbf{#1}} where something describes what the markup means mathematically, as opposed to what it looks like.

• Your current layout is not consistent with the plain page style. You should modify one or other or both.

  • Here's the layout visually:

  

  You can see that you have no space here for a header or footer (or marginal notes). So page numbers etc. simply get cut-off and, indeed, geometry warns you

Package geometry Warning: The marginal notes overrun the paper.
     Add 25.54726pt and more to the right margin.

*geometry* verbose mode - [ preamble ] result:
* driver: pdftex
* paper: a4paper
* layout: <same size as paper>
* layoutoffset:(h,v)=(0.0pt,0.0pt)
* modes: showframe 
* h-part:(L,W,R)=(28.45274pt, 540.60239pt, 28.45274pt)
* v-part:(T,H,B)=(28.45274pt, 788.14136pt, 28.45274pt)
* \paperwidth=597.50787pt
* \paperheight=845.04684pt
* \textwidth=540.60239pt
* \textheight=788.14136pt
* \oddsidemargin=-43.81725pt
* \evensidemargin=-43.81725pt
* \topmargin=-80.81725pt
* \headheight=12.0pt
* \headsep=25.0pt
* \topskip=12.0pt
* \footskip=30.0pt
* \marginparwidth=44.0pt
* \marginparsep=10.0pt
* \columnsep=10.0pt
* \skip\footins=10.8pt plus 4.0pt minus 2.0pt
* \hoffset=0.0pt
* \voffset=0.0pt
* \mag=1000
* \@twocolumnfalse
* \@twosidefalse
* \@mparswitchfalse
* \@reversemarginfalse
* (1in=72.27pt=25.4mm, 1cm=28.453pt)

• 1cm margins and zeroed skips make for fairly ugly results.

  

  This is justifiable if

  1. it is an institutional requirement - institutions are known for demanding aesthetically abhorrent typography, especially, for some reason, North American graduate schools;
  2. this is a crib sheet for a 1-page-cheat-sheet-allowed exam, with a requirement for 1.3 line-spacing, 12pt and 1cm margins, but no requirement for acceptable typography;
  3. it is only for the draft, which only you will read and you don't care/like it.

  If none of these apply, you should reconsider the layout and skips. This is especially true if you are doing this to get your submission into the permitted page count, because making your paper difficult for the reviewers to read isn't usually the best way of achieving that.

  If you really need these settings, you should not use ignoreall, but should configure the layout settings so that things fit on the page. If you don't need marginal notes, for example, marginparwidth=0pt,marginparsep=0pt might be suitable. And similarly for the header and footer areas.

\usepackage[marginparwidth=0pt,marginparsep=0pt,headheight=0pt,headsep=0pt,footskip=0pt,a4paper,margin=1cm]{geometry}

This says you have no header or footer and no marginal notes. This sanitises the settings, so eliminating inconsistencies. It obviously does not ameliorate the typographic mischief.