Linear Algebra II HT23, Determinants
Determinants
When building up determinants as a determinantal mapping $D: M _ n(\mathbb{R}) \to \mathbb{R}$, @State the three conditions a determinantal map must satisfy.
- Multilinear in the columns
- Alternating
- $D(I _ n) = 1$ for the $n \times n$ identity matrix
What’s the geometric interpretation behind a determinantal map being multilinear in the columns, i.e. $D[\ldots,\lambda \pmb{b} _ i + \mu\pmb{c} _ i, \ldots] = \lambda D[\ldots,\pmb{b} _ i,\ldots] + \mu D[\ldots,\pmb{c} _ i,\ldots]$?
It represents the fact that stretching a face will scale the area/volume accordingly.
What’s the geometric interpretation behind a determinantal map being alternating, i.e. $D[\ldots,\pmb{a} _ i,\pmb{a} _ {i+1},\ldots] = 0 \text{ when } \pmb{a} _ i = \pmb{a} _ {i+1}$ ?
A shape with thickness has no volume.
If $\lambda = \det [\ldots, \pmb{a}, \ldots, \pmb{b}, \ldots]$, then what is $\det [\ldots, \pmb{b},\ldots, \pmb{a}, \ldots]$?
When proving that there exists a map $D _ n$ that satisfies the properties of being a determinantal map on $n\times n$ matrices
Multilinear in the columns
\[D[\ldots,\lambda \pmb{b} _ i + \mu\pmb{c} _ i, \ldots] = \lambda D[\ldots,\pmb{b} _ i,\ldots] + \mu D[\ldots,\pmb{c} _ i,\ldots] \]
Alternating
\[ D[\ldots,\pmb{a} _ i,\pmb{a} _ {i+1},\ldots] = 0 \text{ when } \pmb{a} _ i = \pmb{a} _ {i+1} \]
$D(I _ n) = 1$ for the $n \times n$ identity matrix
In the inductive step, what do you define $D _ n$ as, in terms of $D _ {n-1}$?
Can you give the Laplace expansion formula for $\det A$ along row $i$?
@State the Laplace expansion formula for $\det A$ along column $j$.
@State the permutation formula for $\det A$.
Why is $\det A$ multilinear in rows as well as columns, despite this not being in the definition?
Because $\det A = \det A^\intercal$.
Using the Rule of Sarrus (cool name) mnemonic, what is
\[\left \vert \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}\right \vert \]
?
If $A$ is an upper or lower triangular matrix, what is the determinant?
To prove the more general $\det AB = \det A \det B$, what do you show first?
where $E$ is an elementary matrix.
How does the determinant change when you swap two rows of a matrix?
It switches sign.
How does the determinant change when you add a scalar multiple of one row to another in a matrix?
It does not change.
How does the determinant change when you multiply one row of a matrix by a non-zero scalar $\lambda$?
The determinant is multiplied by $\lambda$.