Proposition 8.3.1.
If \(A\) is an \(n \times n\) matrix, then \(\det A^{T} = \det A\text{.}\)
Section 8.3 Properties of Determinants
Subsection 8.3.1 Determinants and Composition
Before talking about composition, there is a property of determinant that follows from the cofactor expansion. I’ll just state it here.
Now I can talk about composition. Algebra considers sets with structure. On the set \(M_n(\RR)\) of \(n \times n\) matrices, the determinant is a new algebraic structure. I would like to investigate how it interacts with existing structures, starting with matrix multiplication.
To compose two transformations, I multiply the matrices. In this composition, the effects on size should compound. If the first transformation doubles size and the second triples it, the composition should multiply the size by a factor of \(6\text{.}\) Similarly, with orientation: if both preserve orientation, the composition should as well. If both reverse orientation, reversing twice should return to the normal orientation. So, two positive or two negatives should result in a positive, and one positive and one negative should result in a negative. These two observations (compounding effect on sze, reversing orientation) are captured in this proposition.
Proposition 8.3.2.
Let \(A\) and \(B\) be two \(n \times n\) matrices. Then \(\det (AB) = \det (A) \det (B)\text{.}\)
Here is a similar proposition about the properties of determinants.
Proposition 8.3.3.
If \(A\) is an invertible \(n \times n\) matrix, then \(\det A^{-1} = \frac{1}{\det A}\text{.}\)
In particular, for \(A\) to be invertible, it must have a non-zero determinant. It turns out this property is sufficient as well as necessary. A non-zero determinant preserves the dimension: it may increase or decrease size, but it doesn’t entirely destroy it. Therefore, there is always a way to go back and to reverse the process.
Proposition 8.3.4.
A \(n \times n\) matrix \(A\) is invertible if and only if \(\det A \neq 0\text{.}\)
I can expand the list of criteria for invertibility by including this new determinant condition. Here is the list that I previously presented in Section 7.4 with the new condition added at the end.
Proposition 8.3.5.
Let \(A\) be an \(n \times n\) matrix. All of the following properties are equivalent.
- \(A\) is invertible.
- \(A\) has rank \(n\text{.}\)
- \(A\) row reduces to the identity matrix.
- \(\Ker(A) = \{0\}\text{.}\)
- \(\Image(A) = \RR^n\text{.}\)
- The columnspace of \(A\) is \(\RR^n\text{.}\)
- The columns of \(A\) are linearly independent.
- The rowspace of \(A\) is \(\RR^n\text{.}\)
- The rows of \(A\) are linearly independent.
- \(Au = v\) has a unique solution \(u\) for any choice of \(v \in \RR^n\text{.}\)
- \(\det A \neq 0\text{.}\)
Subsection 8.3.2 Other Algebraic Properties
Here are two more properties of determinants.
Proposition 8.3.6.
Let \(A\) and \(B\) be \(n \times n\) matrices and let \(\alpha \in \RR\) be a scalar.
\begin{align*}
\det (AB) \amp = \det (BA)\\
\det \alpha A \amp = \alpha^n \det A
\end{align*}
Row reduction is an important operation on matrices. It is convenient to know how the determinant changes under row operations.
Proposition 8.3.7.
Let \(A\) be an \(n \times n\) matrix. If two rows are exchanged, the determinant changes sign (positive to negative, negative to positive). If a row is multiplied by a constant \(\alpha\text{,}\) the determinant is multiplied by the same constant \(\alpha\text{.}\) If a multiple of one row is added to another, the determinant is (surprisingly) unchanged.
All of the properties of determinant listed so far have been multiplicative. The situation for matrix addition and determinants is less elegant: \(\det (A + B)\) has no pleasant identity. This is an interesting contrast from many of the other things in this course: determinants are not linear functions \(M_n(\RR) \rightarrow \RR\) since they do not act nicely with addition. Instead, they act nicely with multiplication. On sets with various algebraic structures, it is common for an operation to interact well with one structure but poorly with the others. It is not surprisingly that the determinant doesn’t work well with matrix multiplication based on the original observation that matrix multiplication doesn’t have a clear geometric meaning. The determinant is a very geometric definition, so an operation that is simply algebraic without a geometric meaning is unlikely to easily cooperate with a strongly geometric construction.
