Matrices: Determinant and Inverse, Step by Step

A matrix is a grid of numbers, and two operations on 2×2 matrices unlock most of linear algebra: the determinant (one number that summarizes the matrix) and the inverse (the matrix that undoes it). Both have short, reliable methods.

The determinant: $ad-bc$

For the matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the determinant is:

$$\det =ad-bc$$

Worked example: for $\left(\begin{array}{cc}2& 3\\ 1& 4\end{array}\right)$:

$$\det =(2\times 4)-(3\times 1)=8-3=5$$

The full formula with more examples is on the determinant of a 2×2 matrix page. One fact worth remembering: if the determinant is zero, the matrix has no inverse.

The inverse: swap, negate, divide

For the same matrix, the inverse is:

$$A^{-1}=\frac{1}{\det }\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$$

In words: swap $a$ and $d$, negate $b$ and $c$, then divide everything by the determinant.

Worked example: for $\left(\begin{array}{cc}2& 3\\ 1& 4\end{array}\right)$ with $\det =5$:

$$A^{-1}=\frac{1}{5}\left(\begin{array}{cc}4& -3\\ -1& 2\end{array}\right)$$

Check by multiplying $A\times A^{-1}$: the result is the identity matrix $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, which confirms the inverse is right. The inverse of a 2×2 matrix page shows the same method with more examples.

Common mistakes

• Subtracting the wrong way round in the determinant — it is $ad-bc$, not $bc-ad$.
• Forgetting to divide by the determinant after swapping and negating.
• Negating the wrong entries — negate $b$ and $c$ (the off-diagonal), never $a$ and $d$.
• Trying to invert a matrix with determinant zero. It has no inverse; stop and check the determinant first.

Practice it

Want to check your work? Try the Matrix Calculator — determinant, inverse, transpose and more, all with steps. Then try similar problems on the practice page.