Differential Equations and Linear Algebra, 4th Edition

Determinants and Invertibility

We began Section 3.6 with the remark that a $2\times 2$ matrix A is invertible if and only if its determinant is nonzero: $\left|\mathbf{\text{A}}\right|\ne 0.$ Now we want to show that this result also holds for $n\times n$ matrices. This connection between determinants and invertibility is closely related to the fact that the determinant function “respects” matrix multiplication in the sense that

$$\left|\mathbf{\text{A}}\mathbf{\text{B}}\right|=\left|\mathbf{\text{A}}\right|\left|\mathbf{\text{B}}\right|$$ (9)

if A and B are $n\times n$ matrices. Our first step is to show that Eq. (9) holds if A is an elementary matrix obtained from the $n\times n$ identity matrix I by performing a single elementary row operation.

LEMMA Multiplication by an Elementary Matrix

If B is an $n\times n$ matrix and E is an $n\times n$ elementary matrix then

$$\left|\mathbf{\text{E}}\mathbf{\text{B}}\right|=\left|\mathbf{\text{E}}\right|\left|\mathbf{\text{B}}\right|.$$ (10)

Proof:

This is really just a restatement ...