Determinant and Inverse of a 2x2 Matrix

Calculates the determinant of the matrix [[a, b], [c, d]] as ad − bc and the four entries of its inverse. When the determinant is zero there is no inverse.

Explanation

For a two by two matrix both the determinant and the inverse have short closed forms, so there is no need for elimination or any general algorithm.

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad \det A = ad - bc

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

$a$ , $b$ , $c$ , $d$ — the four entries: $a$ and $b$ on the top row, $c$ and $d$ on the bottom
$\det A$ — the determinant, $ad - bc$
$A^{-1}$ — the inverse, the matrix that multiplies with $A$ to give the identity

To build the inverse: swap $a$ and $d$ , flip the signs of $b$ and $c$ , then divide everything by the determinant.

Example

Take $a = 4$ , $b = 7$ , $c = 2$ , $d = 6$ . The determinant is

\det A = 4 \times 6 - 7 \times 2 = 24 - 14 = 10

It is not zero, so an inverse exists:

A^{-1} = \dfrac{1}{10} \begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{pmatrix}

Notes

A determinant of 0 means there is no inverse, since the formula would divide by zero. The calculator reports the matrix as singular.
The determinant vanishes exactly when the two rows, or equivalently the two columns, are parallel, one being a multiple of the other.
The absolute value of the determinant is the area of the parallelogram spanned by the two rows. Zero area means that parallelogram has collapsed onto a line.
To check your work, multiply $A$ by $A^{-1}$ and confirm you land on the identity matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ .