Inverse of Invertible 2 x 2 Real Square Matrix

Theorem
Let $\mathbf A$ be an invertible $2 \times 2$ real square matrix defined as:


 * $\mathbf A = \begin{pmatrix}

a & b \\ c & d \end{pmatrix}$

Then its inverse matrix $\mathbf A^{-1}$ is:


 * $\mathbf A^{-1} = \dfrac {1} {\map \det {\mathbf A}} \begin {pmatrix}

d & -b \\ -c & a \end {pmatrix}$

Proof
We construct $\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix}$:


 * $\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix} = \paren {\begin {array} {cc|cc}

a & b & 1 & 0 \\ c & d & 0 & 1 \\ \end {array} }$

In the following, $\sequence {e_n}_{n \mathop \ge 1}$ denotes the sequence of elementary row operations that are to be applied to $\begin {pmatrix} \mathbf A & \mathbf I \end {pmatrix}$.

The matrix that results from having applied $e_1$ to $e_k$ in order is denoted $\begin {pmatrix} \mathbf A_k & \mathbf B_k \end {pmatrix}$.

$e_1 := r_1 \to \dfrac 1 a r_1$

Hence:
 * $\begin {pmatrix} \mathbf A_1 & \mathbf B_1 \end {pmatrix} = \paren {\begin {array} {cc|cc}

1 & \dfrac b a & \dfrac 1 a & 0 \\ c & d & 0 & 1 \\ \end {array} }$

$e_2 := r_2 \to -cr_1 + r_2$


 * $\begin {pmatrix} \mathbf A_2 & \mathbf B_2 \end {pmatrix} = \paren {\begin {array} {cc|cc}

1 & \dfrac b a & \dfrac 1 a & 0 \\ 0 & \dfrac {ad-bc} a & -\dfrac c a & 1 \\ \end {array} }$

$e_3 := r_2 \to \dfrac a {ad-bc} r_1$


 * $\begin {pmatrix} \mathbf A_3 & \mathbf B_3 \end {pmatrix} = \paren {\begin {array} {cc|cc}

1 & \dfrac b a & \dfrac 1 a & 0 \\ 0 & 1 & -\dfrac c {ad-bc} & \dfrac a {ad-bc} \\ \end {array} }$

$e_4 := r_1 \to -\dfrac b a r_2 + r_1$


 * $\begin {pmatrix} \mathbf A_4 & \mathbf B_4 \end {pmatrix} = \paren {\begin {array} {cc|cc}

1 & 0 & \dfrac d {ad-bc} & -\dfrac b {ad-bc} \\ 0 & 1 & -\dfrac c {ad-bc} & \dfrac a {ad-bc} \\ \end {array} }$

and it is seen that $\begin {pmatrix} \mathbf A_4 & \mathbf B_4 \end {pmatrix}$ is the required reduced echelon form:
 * $\mathbf A_4 = \mathbf I$

and so by the Matrix Inverse Algorithm:

Hence the result.

Also see

 * Matrix Product with Adjugate Matrix