Linear Algebra with Applications, 10th Edition

2.1 The Determinant of a Matrix

With each $n\times n$ matrix A, it is possible to associate a scalar, det(A), whose value will tell us whether the matrix is nonsingular. Before proceeding to the general definition, let us consider the following cases.

Case 1. $\mathbf{1}\mathbf{\times }\mathbf{1}$ Matrices If $A=(a)$ is a $1\times 1$ matrix, then A will have a multiplicative inverse if and only if $a\ne 0$. Thus, if we define

$$\text{det}(A)=a$$

then A will be nonsingular if and only if det$(A)\ne 0$.

Case 2. $\mathbf{2}\mathbf{\times }\mathbf{2}$ Matrices Let

$$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$$

By Theorem 1.5.2, A will be nonsingular if and only if it is row equivalent to I. Then, if $a_{11}\ne 0$, we can test whether A is row equivalent to I by performing the following operations:

1. Multiply the second row of A by $a_{11}$:

   $$\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{11}{a}_{21}& a_{11}{a}_{22}\end{array}\right]$$

2. Subtract $a_{21}$ times the first ...