- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
3.8 $\quad$ The determinant of the inverse of a nonsingular matrix
$\quad$ We recall that a square matrix $A$ is called nonsingular if there exists matrix $B,$ called the left inverse of $A,$ such that $BA = I.$ If such a matrix exists, then it is unique, and is also a right inverse such that $AB = I.$ The matrix $B$ is called the inverse of $A$ and is denoted by $A^{-1}.$ The relation between $\det A$ and $\det A^{-1}$ is as follows:
$\quad$ Theorem 3.5. $\quad$ If a matrix $A$ is nonsingular, then $\det A \neq 0$ and we have \begin{align*} \det A^{-1} &= \frac{1}{\det A} \end{align*}
$\quad$ Proof. $\quad$ From the product formula for determinants, we know that \begin{align*} (\det A^{-1})(\det A) &= \det (A^{-1}A) = \det I = 1. \end{align*} Hence, $\det A^{-1} \neq 0$ and $\det A^{-1} = 1/\det A. \quad \blacksquare$
$\quad$ This shows that a nonvanishing determinant is a necessary condition of $A$ being nonsingular. Later, we will show that this condition is also sufficient. That is, $\det A \neq 0$ implies $A$ is nonsingular.