2.6.12 · D5Matrices & Determinants — Introduction
Question bank — Solving systems using matrix inversion
This page hunts the misconceptions, not the arithmetic. Every item is a one-line reveal: read the prompt, commit to an answer out loud, then check. If your reason differs from the reveal even when your verdict matches — that gap is the lesson.
Parent: matrix inversion method. Prerequisite gates: Determinants, Adjoint and Inverse of a Matrix, Consistency of Linear Systems.
True or false — justify
Every statement here is either subtly true or subtly false. The reveal gives the reason, which is the point.
The equation is just the original system of equations re-written, not a new object.
True — multiplying row of by the column reproduces the left side of equation , and equating it to gives equation back exactly. It is repackaging, not approximation.
If exists, then is one valid solution but there may be others.
False — when exists the solution is unique. If and , left-multiply by to get , so no second solution can hide.
guarantees a solution exists; guarantees no solution exists.
The first half is true, the second is false. means no unique solution — it could still have infinitely many, depending on whether (the zero column).
For a matrix, "swap the diagonal and negate the off-diagonal" gives the inverse directly.
False — that operation gives the adjugate, not the inverse. You still must divide by ; skipping the division leaves you off by a scalar factor.
and describe the same system.
False — matrix multiplication is not commutative, and for a column vector the product is usually not even defined. Order and side both carry meaning.
If a system has infinitely many solutions, its coefficient matrix must have .
True — a non-zero determinant forces a unique solution, so infinitely many solutions can only occur when the inverse fails to exist, i.e. .
The identity holds even when .
True — it is an algebraic identity valid for every square matrix. When it simply says (the zero matrix), which is what powers the consistency test.
Two different linear systems can share the same coefficient matrix but have different solvability.
True — same means same , so both are unique-or-both-degenerate; but if , the value of decides "no solution" versus "infinitely many". Solvability is not determined by alone in the degenerate case.
Spot the error
Each line contains a claimed step or fact. Find what is wrong and why.
"From , divide both sides by to get ."
There is no matrix division. The legal move is left-multiply by , giving ; the notation is meaningless because it hides which side the inverse sits on.
"Since , we left-multiply by to get ."
The verdict "left-multiply" is right but the result is wrong: left-multiplying gives , with the inverse on the left. would be right-multiplication and may not even be dimensionally valid.
", so I'll compute and press on carefully."
You cannot divide by zero — does not exist here. The method is over; you must switch to consistency analysis using .
" is the matrix of cofactors placed in their natural positions."
Missing the transpose: . The adjugate is the transpose of the cofactor matrix; forgetting it silently corrupts every entry off the diagonal.
"For the , the inverse is ."
The diagonal entries must be swapped: it should be . Keeping and in place fails the check .
"I found , so I don't need to substitute back — the algebra is exact."
The algebra is exact only if you computed correctly; a single sign slip in adj or det is invisible until you substitute. The check catches those slips cheaply.
" and , therefore the system definitely has infinitely many solutions."
The reveal is a maybe: removes the "no solution" verdict but does not by itself prove existence — you must check by substitution, since it can still be inconsistent in some configurations.
Why the consistency test is necessary but not sufficient — a visual walk-through
Why questions
Answer the reason, not just the fact.
Why do we left-multiply by rather than right-multiply?
Because sits on the left of in , and only left-multiplication produces adjacent to ; right-multiplying would strand the inverse on the wrong side of .
Why does act as the gate for a unique solution?
Because requires dividing by ; geometrically means the unit square maps to a parallelogram of non-zero area (space isn't flattened), so distinct inputs stay distinct and can be reversed.
Why can't a "flattening" machine () be undone uniquely?
Flattening maps a whole line (or plane) of different inputs onto the same output, so given the output you cannot recover which input produced it — the reverse machine has no single answer.
Why does the identity explain the consistency test?
Multiplying by gives ; when the left side is , forcing , so any violation of that equality proves no exists.
Why is matrix inversion called "mistake-proof" compared to elimination?
The steps are a fixed mechanical recipe — det, adjugate, multiply — with no branching human choices about which row to subtract, so the same input always yields the same procedure and a built-in check .
Why must be square for to make sense?
Only square matrices can have a two-sided inverse satisfying ; a non-square describes a different number of equations than unknowns and cannot be uniquely reversed this way.
Why does dividing by appear, geometrically, as a scaling?
is the factor by which stretches area/volume; to reverse the transformation the inverse must shrink by that same factor, which is precisely the out front.
Edge cases
The boundary and degenerate scenarios the method invites. The figure gives concrete pictures for the two homogeneous cases so the statements aren't just words.
(all constants zero, the zero column) and : what is ?
The unique solution is — the trivial solution. A homogeneous system with an invertible coefficient matrix forces every variable to zero (left panel of the figure: two lines crossing only at the origin).
and : what changes?
Now automatically, so the "no solution" branch is impossible; a homogeneous singular system always has infinitely many solutions — the whole overlapping line through the origin (right panel of the figure).
What is when , the identity machine?
itself, since ; the machine that changes nothing is undone by doing nothing, and then .
If two rows of are identical, can you invert?
No — identical rows force , so the machine flattens space and no inverse exists; you must use consistency analysis instead.
A single equation in a single unknown, : does the method reduce to ordinary algebra?
Yes — here , , and , valid exactly when , which mirrors "can't divide by zero" for the case.
is very small but non-zero (say ): is the method still valid?
Mathematically yes — a unique solution exists and is exact — but the factor is huge, so tiny errors in get amplified enormously (the system is "nearly" un-invertible).
If exists, is ?
Yes — undoing the reverse machine restores the original machine, and algebraically says is precisely the inverse of .
Can ever coexist with a unique solution?
No — means the inverse fails, and a unique solution would let you reconstruct , a contradiction. Zero determinant always means "either none or infinitely many."
Connections
- Determinants — the gate every trap circles back to.
- Adjoint and Inverse of a Matrix — where the transpose-the-cofactors slip lives.
- Consistency of Linear Systems — the branch and test.
- Cramer's Rule — shares the same gate, another lens on uniqueness.
- Cofactors and Minors — the pieces of .