5.1 Properties of Determinants
1. The determinant of the $nxn$ identity matrix is 1.
\[ \left\vert \begin{array}{cc}1&0\\0&1 \end{array}\right\vert = 1\]
and
\[ \left\vert \begin{array}{cccc}1&0&0&...&0\\0&1&..\\.&..\\0&0&0&...&1 \end{array}\right\vert = 1\]
2. The determinant changes sign when 2 rows are exchanged (sign reversal):
Check:
\[ \left\vert \begin{array}{cc}c&d\\a&b \end{array}\right\vert = - \left\vert \begin{array}{cc}a&b\\c&d \end{array}\right\vert\]
3. The determinant is a linear function of each row separately (all other rows stay fixed).
Multiply row 1 by t:
\[ \left\vert \begin{array}{cc}ta&tb\\c&d \end{array}\right\vert = t\left\vert\begin{array}{cc}a&b\\c&d \end{array}\right\vert \]
Add row 1 of A to row 1 of A':
\[ \left\vert \begin{array}{cc}a+a'&b+b'\\c&d \end{array}\right\vert = \left\vert\begin{array}{cc}a&b\\c&d \end{array}\right\vert + \left\vert\begin{array}{cc}a'&b'\\c&d \end{array}\right\vert\]
4. If 2 rows of A are equal, then det A = 0.
Check
\[ \left\vert \begin{array}{cc}a&b\\a&b \end{array}\right\vert = 0 \]
5. Subtracting a multiple of 1 row from another row leaves det A unchanged.
By rule 2 & rule 4.
\[ \left\vert \begin{array}{cc}a&b\\c-la&d-lb \end{array}\right\vert = \left\vert\begin{array}{cc}a&b\\c&d \end{array}\right\vert \]
6. A matrix with a row of zeros has det A=0.
\[ \left\vert \begin{array}{cc}a&b\\0&0 \end{array}\right\vert = 0 \]
7. If A is triangular, then det A = $a_{11}a_{12}\cdots a_{nn}$ = product of diagonal entries.
\[ \left\vert \begin{array}{cc}a&b\\0&d \end{array}\right\vert = \left\vert \begin{array}{cc}a&0\\c&d \end{array}\right\vert = ad \]
8. If A is singular then det A$=0$. If A is invertible then det A $\neq 0$
$\left\vert \begin{array}{cc}a&b\\c&d \end{array}\right\vert $ is singular iff $ad-bc=0$.
9. The determinant of AB equals det A x det B : |AB|=|A||B|
\[ \left\vert \begin{array}{cc}a&b\\c&d \end{array}\right\vert \left\vert \begin{array}{cc}p&q\\r&s \end{array}\right\vert = \left\vert \begin{array}{cc}aq+br&aq+bs\\cp+dr&cq+ds \end{array}\right\vert \]
10. The transpose $A^T$ has the same determinant as $A$.
\[ \left\vert \begin{array}{cc}a&b\\c&d \end{array}\right\vert = \left\vert \begin{array}{cc}a&c\\b&d \end{array}\right\vert \]
Key Ideas
i. The determinant is defined by $\text{det } I=1$, sign reversal and linearity in each row.
ii. After elimination $\text{det A}$ is $\pm$ (product of the pivots).
iii. The determinant is zero exactly when $A$ is not invertible.
iv. Two remarkable properties are $\text{det }AB = \text{(det }A\text{)(det B})$ and $\text{det A}^T = \text{det A}$.
5.2 Permutations and Cofactors
The determinant can be found in 3 ways:
I. Pivots II. Big formula III. Cofactors
I. Pivots method
Using elimination & row exchanges, convert $A$ to $LU$. The permutation matrix $P$ from $PA=LU$ has determinant $\pm$. The determinant is $\pm \times \text{(product of the pivots)}$
$\text{(det P)(det A)=(det L)(det U)}$ gives $\text{det A} = \pm(d_1d_2\cdots d_n)$
II. Big Formula method
The formula has $n!$ terms
In the big formula method each product has 1 entry from each row and 1 entry from each column. These can be obtained via a permutation matrix. The determinant is the sum of these $n!$ determinants multiplied by $\pm1$ depending upon whether the permutation matrix $P$ is even or odd. Thus we have $\text{det }A$ = sum over all $n!$ column permutations $P=( \alpha,\beta \cdots \omega) $ = \[\sum(\text{det }P)a_{1\alpha}a_{2\beta} \cdots a_{n\omega} = \sum\pm a_{1\alpha}a_{2\beta} \cdots a_{n\omega} \]
III. Cofactor formula
The determinant is the dot product of any row $i$ of $A$ with its cofactors using other rows: \[\text{det }A = a_{i1}C_{i1}+a_{i2}C_{i2} + \cdots + a_{in}C_{in} \]
Each cofactor $C_{ij}$ (order $n-1$, without row $i$ and column $j$) includes its correct sign: $C_{ij} = (-1)^{i+j} \text{det }M_{ij}$
5.3 Cramer's Rule, Inverses and Volumes
I. Cramer's Rule
If $\text{det A}$ is not zero, $Ax=b$ has the unique solution
\[ x_1 = \frac{\text{det }B_1}{\text{det }A} \hspace{10pt} x_2 = \frac{\text{det }B_2}{\text{det }A} \hspace{10pt} \cdots \hspace{10pt} x_n = \frac{\text{det }B_n}{\text{det }A} \]
The matrix $B_j$ has the $jth$ column of $A$ replaced by the vector $b$.
II. Formula for $A^{-1}$
The $i,j$ entry of $A^{-1}$ is the cofactor $C_{ji}$ (not $C_{ij}$) divided by $\text{det }A$:
\[ (A^{-1})_{ij} =\frac{C_{ji}}{\text{det }A} \hspace{10pt} \text{and} \hspace{10pt} A^{-1} = \frac{C^T}{\text{det }A} \]
III. Area of triangle
The triangle with corners $(x_1,y_1)$ and $(x_2,y_2)$ and $(x_3,y_3)$ has area=$\frac{1}{2}$(determinant):
\[ \text{Area } = \frac{1}{2} \left\vert \begin{array}{ccc}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1 \end{array}\right\vert \]
when $(x_3,y_3)=(0,0)$
\[ \text{Area } = \frac{1}{2} \left\vert \begin{array}{cc}x_1&y_1\\x_2&y_2 \end{array}\right\vert \]
IV. Cross Product
The cross product of $u$ = $(u_1,u_2,u_3)$ and $v$=$(v_1,v_2,v_3)$ is the vector
\[ u \text { x } v = \left\vert \begin{array}{ccc}i&j&k\\u_1&u_2&u_3\\v_1&v_2&v_3 \end{array}\right\vert = (u_2v_3-u_3v_2)i + (u_3v_1-u_1v_3)j + (u_1v_2-u_2v_1)k \]
This vector $u \text{ x } v$ is perpendicular to $u$ and $v$. The cross product $v \text{ x } u$ is $-(u \text{ x } v)$.
The length of $u \text{ x } v$ equals the area of the parallelogram with sides $u$ and $v$.
The cross product is a vector with length $||u|||v|| |\text{sin }\theta|$. Its direction is perpendicular to $u$ and $v$.
V. Triple Product
The triple product is defined as $(u \text{ x } v) \cdot w$ = \[ \left\vert \begin{array}{ccc}w_1&w_2&w_3\\u_1&u_2&u_3\\v_1&v_2&v_3 \end{array}\right\vert \]
$(u \text{ x } v) \cdot w$ equals the volume of the box with sides $u$, $v$ and $w$.
$(u \text{ x } v) \cdot w$ = $0$ exactly when the vectors $u$, $v$, $w$ lie in the same plane. |