Inverse solution of a system
Solve a square linear system using an inverse.
Inverse solution of a system is one of 2 systems formulas in the linear algebra section of this library, and it is used at university level.
Why inverse solution of a system works
The system says that A applied to some unknown vector produced b. If A can be undone, applying that undo to b retraces the step and lands back on the unknown. It is the same reasoning as dividing both sides of 5x = 20 by 5, except the undo is a matrix rather than a fraction.
What each symbol means
$A$ is coefficient matrix, $\mathbf x$ unknown vector, and $\mathbf b$ constants.
Inverse solution of a system: when it holds
$A$ must be square and invertible; elimination is usually more efficient computationally.
When it stops applying
If det A = 0 the inverse does not exist and this route is closed, though the system may still have infinitely many solutions. The method also needs A to be square; with more equations than unknowns there is no inverse to find and you turn to least squares.
Inverse solution of a system: a worked example
Verify a solution by multiplying $A\mathbf x$ and checking that it equals $\mathbf b$.
The mistake to avoid
What people do: Students write the answer as b times A⁻¹, putting the inverse on the right.
Why it goes wrong: Matrix products depend on order, and the inverse has to meet A directly to cancel it. With A⁻¹ on the wrong side the dimensions often do not even line up.
Do this instead: Keep the inverse in front: x = A⁻¹b. For A = [[2, 1], [1, 1]] and b = (5, 3), the inverse [[1, −1], [−1, 2]] gives x = (2, 1), and substituting back gives 2(2) + 1 = 5 and 2 + 1 = 3.
Inverse solution of a system: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A$ is coefficient matrix, $\mathbf x$ unknown vector, and $\mathbf b$ constants.
- Check the conditions before substituting. $A$ must be square and invertible; elimination is usually more efficient computationally.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most linear algebra slips.
Where this formula fits
- Subject
- Linear Algebra formulas — 17 entries in this library
- Topic
- Systems
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where inverse solution of a system comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Solving Linear Systems with Matrices — the lesson behind this formula: use row operations to reveal solutions and dependencies.
- Trivial Solution Calculator — check your substitution and the value it produces.
- Study linear algebra — the subject guide that explains the ideas these formulas compress.
- Linear Algebra Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 linear algebra formulas — the full grouped reference, or the complete formula library.
Questions about inverse solution of a system
Why do textbooks say elimination is better?
Because computing a full inverse does roughly three times the arithmetic of solving directly, and it loses more accuracy in floating point. The inverse is a fine idea and usually a poor algorithm.
Does det A = 0 mean there are no solutions?
Not necessarily. It means there is no unique solution, so the truth is either no solution at all or an entire line or plane of them. Row reduction is what tells the two cases apart.
Is it worth finding the inverse if I have many right-hand sides?
Sometimes, but a stored factorization is still better. Once A is factored, each new b costs only a quick forward and backward pass, with better accuracy than multiplying by an inverse.
How do I check a solution without redoing the work?
Multiply A by your answer and compare it with b. That takes one matrix-vector product and confirms the result independently of how you found it.