How Do You Find the Determinant and Inverse of a 2×2 Matrix?

For a $2\times2$ matrix, calculate the determinant as (ad-bc). If it is non-zero, the inverse exists: interchange the diagonal entries, negate the off-diagonal entries, and divide the resulting matrix by the determinant.

This is AHL 1.14, Introduction to matrices, so it is HL-only content in IB Mathematics: Applications and Interpretation. It may be tested on any HL paper, and a GDC is available throughout.

Let

$$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}.$$

Its determinant is

$$\det(A)=ad-bc.$$

The determinant determines whether the matrix transformation can be reversed. If $\det(A)\neq0$, every output corresponds to exactly one input, and

$$A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}.$$

For example, consider

$$A=\begin{pmatrix}3&2\\5&4\end{pmatrix}.$$

First calculate the determinant:

$$\det(A)=(3)(4)-(2)(5)=12-10=2.$$

Since $2\neq0$, the inverse exists:

$$A^{-1}=\frac{1}{2}\begin{pmatrix}4&-2\\-5&3\end{pmatrix}=\begin{pmatrix}2&-1\\-\frac{5}{2}&\frac{3}{2}\end{pmatrix}.$$

Verify by multiplying in either order. A correct inverse satisfies $AA^{-1}=A^{-1}A=I$, where $I$ is the identity matrix.

The inverse also solves matrix equations and corresponding systems of linear equations. If $AX=B$, then $X=A^{-1}B$. Because matrix multiplication is not commutative, multiplication must occur on the correct side. For $XA=B$, instead use . The absolute value of the determinant is also the area scale factor of the associated two-dimensional transformation; a zero determinant collapses area, making reversal impossible.

A common misconception is that every square matrix has an inverse. If $ad-bc=0$, the matrix is singular, so $A^{-1}$ does not exist because the formula requires division by zero.

Exam technique: Show the determinant calculation before stating whether the inverse exists. Write the inverse formula and substitute carefully before using your GDC to verify it. Preserve exact fractions unless decimals are requested, and explicitly state “the inverse does not exist” when the determinant is zero.