Math 1410, Spring 2020
Determinants
| Sean Fitzpatrick |
|---|
| University of Lethbridge |
 |
Recap
Warm-Up
Compute the determinant of the given matrices:
\(A = \bbm 3\amp -2\\5\amp 4\ebm\)
\(B = \bbm 1\amp 3\amp 0\\2\amp -1\amp 4\\0\amp 1\amp -5\ebm\)
Computing Determinants
Minors and Cofactors
Given an \(n\times n\) matrix \(A = [a_{ij}]\text{,}\)
The \((i,j)\) minor of \(A\) is the \((n-1)\times (n-1)\) matrix \(M_{ij}\) obtained by deleting row \(i\) and column \(j\) of \(A\)
The \((i,j)\) cofactor of \(A\) is the number \(C_{ij}\) defined by
\begin{equation*} C_{ij} = (-1)^{i+j}\det M_{ij}\text{.} \end{equation*}
Note: \((-1)^{i+j}\) equals \(+1\) if \(i+j\) is even, and \(-1\) if \(i+j\) is odd.
(Now do some examples, Sean)
The Determinant (\(n\times n\))
Definition:
Let \(A = [a_{ij}]\) be an \(n\times n\) matrix. The determinant of \(A\) is the number \(\det A\) given by
The sum above is called a cofactor expansion.
Note that if \(A\) is \(4\times 4\) or larger, this definition is recursive: each \(C_{ij}\) is a \(3\times 3\) determinant, which must be computed using cofactor expansion in terms of \(2\times 2\) determinants.
Example
Compute the determinant of
Laplace expansion theorem
Theorem:
The determinant can be computed using cofactor expansion along any row.
In fact, it turns out \(\det A = \det A^T\) for any \(n\times n\) matrix \(A\text{,}\) so we can also do cofactor expansion along any column.
Example:
Compute the determinant of \(A = \bbm 2\amp 1\amp -1\amp 3\\0\amp 5\amp 0\amp 0\\3\amp 0\amp 1\amp -2\\4\amp -4\amp 0\amp 1\ebm\)
Determinants and row operations
Triangular matrices
Definition:
A matrix \(A=[a_{ij}]\) is called upper-triangular if \(a_{ij}=0\) whenever \(i\gt j\text{,}\) and lower-triangular if \(a_{ij}=0\) whenever \(i\lt j\text{.}\)
A matrix \(D=[d_{ij}]\) is called diagonal if \(d_{ij}=0\) whenever \(i\neq j\text{.}\)
An upper triangular matrix has all zeros below the main diagonal.
A lower triangular matrix has all zeros above the main diagonal.
A diagonal matrix has zeros both above and below the main diagonal. It also counts as triangular.
Examples
This is a slide containing very little, other than to let us know that Sean is about to write some triangular matrices on the board, and then compute their determinants.
OK, this slide also contains a theorem: if \(A\) is a triangular matrix, then \(\det A\) is given by the product of the diagonal entries of \(A\text{:}\)
\begin{equation*} \det A = a_{11}a_{22}\cdots a_{nn}\text{.} \end{equation*}
Effect of row operations
Moral of the story so far:
Determinants are generally hard to compute. (Well, not so much hard as annoying.)
Except if the matrix is triangular. Then determinants are easy.
We know how to put a matrix into triangular form. (Row echelon form is triangular!)
Looks like we'd better figure out what row operations do to a determinant!
Theorem:
If \(B\) is obtained from \(A\) using the row operation \(R_i\leftrightarrow R_j\text{,}\) then \(\det B = -\det A\text{.}\)
If \(B\) is obtained from \(A\) using the row operation \(kR_i \to R_i\text{,}\) then \(\det B = k\det A\text{.}\)
If \(B\) is obtained from \(A\) using the row operation \(R_i+kR_j\to R_i\text{,}\) then \(\det B=\det A\text{.}\)
Examples
Compute the determinant of:
\(A = \bbm 2\amp 6\amp -4\\-1\amp 2\amp 3\\2\amp 5\amp 1\ebm\)
\(B = \bbm 6\amp -3\amp 1\\-2\amp 1\amp 4\\2\amp 5\amp -3\ebm\) (Is there a more efficient option than finding triangular form?)
\(C = \bbm 1\amp 2\amp 0\amp -5\\0\amp 3\amp -2\amp 1\\-1\amp 3\amp 0\amp -4\\-2\amp 1\amp -3\amp 5\ebm\)
Properties of Determinants
Theorem:
Let \(A\) and \(B\) be \(n\times n\) matrices. Then:
\(\det(AB) = \det(A)\det(B)\)
\(\det(A^T) = \det(A)\)
\(\det(kA)=k^n\det(A)\)
Theorem:
A matrix \(A\) is invertible if and only if \(\det(A)\neq 0\text{.}\) Furthermore, if \(A\) is invertible, then