Matrix Inverse Calculator (2×2)

About Matrix Inverse Calculator (2×2)

The inverse of a 2×2 matrix exists only when the determinant is non-zero. Multiply the adjugate by 1/det.

How to use this calculator

1. Enter the four entries of the \(2\times 2\) matrix in row order.
2. Check the determinant, which is \(ad-bc\).
3. If the determinant is \(0\), the matrix has no inverse.
4. If the determinant is not \(0\), the calculator returns the inverse and shows the steps.

The formula explained

For a \(2\times 2\) matrix \(A=\begin{pmatrix}a & b \ c & d\end{pmatrix}\), the formula finds the matrix that multiplies with \(A\) to give the identity matrix. It works only when \(ad-bc\neq 0\).

• a = top left entry of the matrix
• b = top right entry of the matrix
• c = bottom left entry of the matrix
• d = bottom right entry of the matrix
• ad-bc = the determinant of the matrix
• \(A^{-1}\) = the inverse of matrix \(A\)

Step by step method

1. Write the matrix as \(A=\begin{pmatrix}a & b \ c & d\end{pmatrix}\).
2. Compute the determinant \(ad-bc\). If it is \(0\), stop because the inverse does not exist.
3. Swap \(a\) and \(d\), change the signs of \(b\) and \(c\), then divide every entry by \(ad-bc\).

Worked example

Problem. Find the inverse of \(\begin{pmatrix}2 & 5 \ 1 & 3\end{pmatrix}\).

1. Compute the determinant, \(ad-bc=(2)(3)-(5)(1)=6-5=1\).
2. Swap \(2\) and \(3\), and change the signs of \(5\) and \(1\), giving \(\begin{pmatrix}3 & -5 \ -1 & 2\end{pmatrix}\).
3. Divide each entry by \(1\), so the inverse is \(\begin{pmatrix}3 & -5 \ -1 & 2\end{pmatrix}\).

Tips and common mistakes

• If the determinant is \(0\), there is no inverse, so the calculator will not give one.
• A common mistake is forgetting to change the signs of \(b\) and \(c\) before dividing by the determinant.

Frequently asked questions