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The determinant of a square matrix.

Theorem 10.1.1   A square matrix A is invertible if, and only if, its determinant is not equal to zero.

Example 10.1.2  

Take $A=\begin{pmatrix}1&2\\ 3&4 \end{pmatrix}$. Then: $\vert A\vert=\begin{vmatrix}1&2\\ 3&4 \end{vmatrix}= 1 \times 4 - 2 \times 3 =-2 \neq 0$. The matrix A is invertible.

You can check that $A^{-1}= -\frac 12 \begin{pmatrix}4&-2\\ -3&1 \end{pmatrix}$.

Noah Dana-Picard