Sep 22

Inverse of a 2×2 matrix

By Tutor GuyNo Comments

 

\left ( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right )^{-1} = \text{ ?}

 Instead of calculating the inverse of a 2 x 2 matrix, it’s easier to remember this simple manipulation:

  • Switch the a and d terms, and
  • change the signs on the b and c terms.
  • Then divide every term by the determinant (ad - bc). The result will be the inverse of the original matrix.

In general:

\left ( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right )^{-1} = \dfrac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

 Example:

\left ( \begin{bmatrix} 2 & 1 \\ -3 & 7 \end{bmatrix} \right )^{-1} = \dfrac{1}{2 \cdot 7-1 \cdot (-3)} \begin{bmatrix} 7 & -1 \\ 3 & 2 \end{bmatrix} = \dfrac{1}{17} \begin{bmatrix} 7 & -1 \\ 3 & 2 \end{bmatrix}= \begin{bmatrix} ^7 \! / _{17} & ^{-1} \! / \! _{17} \\ ^3 \! / \! _{17} & ^2 \! / \! _{17} \end{bmatrix}

Precalc/Trig
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