Skill: Calculate the inverse of a 2 by 2 matrix

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For a 2×2 matrix with real entries, the inverse is the matrix that undoes its action: multiplying the matrix by its inverse in either order gives the identity matrix. Its entries are obtained by swapping the diagonal entries, negating the off-diagonal entries, and dividing by the determinant; an inverse exists only when the determinant is nonzero, so a matrix with determinant zero cannot be inverted. This treatment does not extend to larger matrices or more general number systems.

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Detailed Explanation: Calculate the inverse of a 2 by 2 matrix

For a $2\times2$ matrix

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

first calculate its determinant, $ad-bc$ . If the determinant is zero, the matrix has no inverse. Otherwise, swap the diagonal entries, change the signs of the off-diagonal entries, and divide every entry by the determinant:

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

Example: Find the inverse of

A=\begin{pmatrix}2&3\\1&4\end{pmatrix}.

Calculate the determinant:

2(4)-3(1)=5.

Since $5\ne0$ , the inverse exists.

Swap the diagonal entries, $2$ and $4$ , and change the signs of $3$ and $1$ :

\begin{pmatrix}4&-3\\-1&2\end{pmatrix}.

Divide by the determinant:

A^{-1}=\frac15\begin{pmatrix}4&-3\\-1&2\end{pmatrix} =\begin{pmatrix}\frac45&-\frac35\\-\frac15&\frac25\end{pmatrix}.

You can check the result by multiplying $A$ by $A^{-1}$ ; the product is the identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ .

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