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4.1. Determinants of $2\times 2$ matrices

Here is the formula for $2\times 2$ determinants.

Let $A\in\operatorname{M}_{{2}}({{\mathbb{R}}})$ . The determinant of $A$ is the real number

\det A=\left|\begin{array}[]{cc}a&b\\ c&d\end{array}\right|=ad-bc.

Beware the notation: $|a_{ij}|$ is a number, while $(a_{ij})$ and $[a_{ij}]$ are matrices.

Hence, we may reformulate Theorem 3.1.7 as follows:

Theorem 3.1.7 Let $A\in\operatorname{M}_{{2}}({{\mathbb{R}}})$ . Then, $A$ is invertible if and only if $\det A\neq 0$ , in which case,

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

Example 4.1.2.

Let $A=\begin{pmatrix}5&1\\ 4&2\end{pmatrix}$ . Then $\det A=5\cdot 2-1\cdot 4=6,$ so by the above Theorem, $A$ is invertible and ${A}^{-1}=\frac{1}{6}\begin{pmatrix}2&-1\\ -4&5\end{pmatrix}.$

Example 4.1.3.

Let $A=\begin{pmatrix}1&-1\\ -1&1\end{pmatrix}$ . Then $\det A=1\cdot 1-(-1)(-1)=0.$ So $A$ is not invertible.

4.1. Determinants of 2×2 matrices