Two-by-two inverse is one of 5 matrices formulas in the linear algebra section of this library, and it is used at high school · university level.
Why two-by-two inverse works
Swapping the diagonal entries and negating the off-diagonal ones produces a matrix called the adjugate. Multiply the original by it and every off-diagonal term cancels while both diagonal terms come out as ad − bc, so the product is that number times the identity. Dividing by ad − bc is the final step that turns it into the identity itself.
What each symbol means
$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$.
Two-by-two inverse: when it holds
$ad-bc\ne0$.
When it stops applying
When ad − bc is 0 there is nothing to divide by and no inverse exists. That is not a gap in the formula: such a matrix flattens the plane onto a line, and once two different inputs land on the same output, no rule can send that output back.
Two-by-two inverse: a worked example
For $A=\begin{bmatrix}2&1\\1&1\end{bmatrix}$, $A^{-1}=\begin{bmatrix}1&-1\\-1&2\end{bmatrix}$.
The mistake to avoid
What people do: Students negate a and d and swap b and c, which is the pattern turned inside out.
Why it goes wrong: The rule is the other way round: the corners of the main diagonal trade places while keeping their signs, and the other two entries stay put while gaining a minus sign.
Do this instead: For A = [[2, 1], [1, 1]] the determinant is 1, so the inverse is [[1, −1], [−1, 2]]. Multiply the two together and you should land on [[1, 0], [0, 1]], which is the check that catches a scrambled pattern immediately.
Two-by-two inverse: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$.
- Check the conditions before substituting. $ad-bc\ne0$.
- 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
- Matrices
- Level
- High school · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where two-by-two inverse comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Matrix Operations — the lesson behind this formula: add, scale, and multiply matrices with attention to dimensions.
- Linear Algebra 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 two-by-two inverse
Is A⁻¹ the same as 1 divided by A?
No, because dividing by a matrix is not defined. A⁻¹ is the matrix that undoes A, and the only division involved is by the single number ad − bc.
How do I check my inverse is correct?
Multiply it by the original in either order; both products must give the identity. This takes about fifteen seconds and catches sign errors that are otherwise easy to miss.
Does a similar shortcut exist for a 3 by 3?
There is a formula using the adjugate, but it needs nine cofactors and is slow and error-prone by hand. For 3 by 3 and larger, row reduction alongside the identity is the practical method.
What is the inverse of the inverse?
The original matrix. Undoing an undo puts you back where you started, and you can see it in the formula, since applying the swap-and-negate pattern twice restores every entry.