1 Determinants
2 Defining Properties
2.1 Property 1: Determinant of $I$
2.2 Property 2: Sign Reversal
2.3 Property 3: Linearity
3.1 Property 4: Equal Rows
3.2 Property 5: Linear Combinations
3.3 Property 6: Zero Row
3.4 Property 7: Determinant of $U$
3.5 Property 8: Singularity Test
3.6 Property 9: Product Rule
3.7 Property 10: Transposition
4 Calculating Determinants
5 Determinant Formula
5.1 $2 \times 2$ case
5.3 $n \times n$ case: "Big Formula"
6 The Pivot Formula
7 Cofactors
7.1 $3 \times 3$ Case: Intuition
7.2 Cofactors
7.3 The Cofactor Formula
8.1 Cramer's Rule
8.2 Volume
8.3 Area of Triangle
A determinant is a value associated with a square matrix $A$
it provides important information about invertability of the matrix
it's denoted as $\text{det } A$ or sometimes $| A |$
Defining Properties
These properties define what a determinant is (they don't say how to compute it)
Property 1: Determinant of $I$
$\text{det } I = 1$
Property 2: Sign Reversal
let $A'$ be a matrix $A$ with two rows exchanged, then $\text{det } A' = \text{det } A$
Consequence of property 2:
for a Permutation Matrix $P$
$\text{det } P = 1$ if if has even number of row exchanges
and $\text{det } P = -1$ is it has odd number of exchanges
Property 3: Linearity
determinant is a linear function of a row - if all other rows stays the same
3a) $\begin{vmatrix}
t a_{11} & t a_{12} & t a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{vmatrix} = t \cdot \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{vmatrix}$
3b) $\begin{vmatrix}
a_{11} + a'_{11} & a_{12} + a'_{12} & a_{13} + a'_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{vmatrix} = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{vmatrix} + \begin{vmatrix} a'_{11} & a'_{12} & a'_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{vmatrix}$
applies to all the rows: we always can put any row to the first position (property 2), then apply the property 3, and then put the row back
These properties are consequences of the defining properties
Property 4: Equal Rows
if 2 rows are equal, then $\text{det } A = 0$
proof: if we exchange two rows, nothing happens to the matrix, but property 2 says the sign should be reversed
Property 5: Linear Combinations
if we subtract a multiple of row $i$ from the row $k$, the determinant remains the same
for Gaussian Elimination it means that $\text{det } A = \text{det } U$
$\begin{vmatrix}
a_{11} & a_{12} \\ a_{21} - c a_{11} & a_{22} - c a_{12} \\ \end{vmatrix} \ \mathop{=}\limits^{3^{\circ}} \ \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix} - c \begin{vmatrix} a_{11} & a_{12} \\ a_{11} & a_{12} \\ \end{vmatrix} \ \mathop{=}\limits^{4^{\circ}} \ \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix} - c 0 = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix}$
Property 6: Zero Row
if we have a rows full of zeros, then $\text{det } A = 0$
$c \cdot \begin{vmatrix}
a_{11} & a_{12} \\ 0 & 0 \\ \end{vmatrix} = \begin{vmatrix} a_{11} & a_{12} \\ c \cdot 0 & c \cdot 0 \\ \end{vmatrix} = \begin{vmatrix} a_{11} & a_{12} \\ 0 & 0 \\ \end{vmatrix}$
the only possible way for this to be valid is when the det is 0
Property 7: Determinant of $U$
for upper-triangular matrix $U$, determinant of $U$ is the product of elements on the main diagonal
d_1 & 0 & 0 \\ a_{21} & d_2 & 0 \\ a_{31} & a_{32} & d_3 \\ \end{vmatrix} = \prod d_i$
if we can do $LU$ factorization, then we can do $LDU$ factorization as well
and by property 5, $\text{det } A = \text{det } U = \text{det } D$
d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \\ \end{vmatrix} \ \mathop{=}\limits^{3^{\circ}} \ d_1 \cdot \begin{vmatrix} 1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \\ \end{vmatrix} = d_1 d_2 \cdot \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & d_3 \\ \end{vmatrix} = d_1 d_2 d_3 \cdot \text{det } I = \prod d_i$
the easiest way to compute the determinant is to apply $A = LU$ Factorization and then compute $\text{det } U = \prod_i d_i$
note that you should be careful with row exchanges!
what if some $d_i = 0$? then $\text{det } U = 0$
Property 8: Singularity Test
when $A$ is singular, then $\text{det } A = 0$
when $A$ is non-singular, then $\text{det } A \ne 0$
it makes it a good test for invertability
directly follows from property 7
compute $A = LU$ factorization
if the matrix is singular, then at least one $d_i = 0$, then $\text{det } U = 0$
if $A$ is not singular, then no pivot is 0, thus $\text{det } U \ne 0$
Property 9: Product Rule
$\text{det } AB = \text{det } A \cdot \text{det } B$
consider ratio $D(A) = \cfrac{\text{det } AB}{\text{det } B}$ (for $\text{det } B \ne 0$)
note that $D(A)$ obeys the properties 1, 2, 3 of $\text{det } A$
property 1: if $A = I$, then $D(A) = \cfrac{\text{det } IB}{\text{det } B} = 1$
property 2: if we exchange two rows of $A$, the same rows are exchanged for $AB$, thus $\text{det } AB$ changes the sign, and so does $D(A)$
property 3:
3a) multiply row 1 of $A$ by $c$, then row 1 of $AB$ also gets multiplied by $c$
3b) add $[a'_{11}, \ ... \ , a'_{1n}]$ to row 1 of $A$ - then row 1 of $AB$ gets row 1 of $A' B$ (where $A'$ is $A$ with row 1 replaced)
illustration: $\text{det } \begin{bmatrix}
a_{11} + a'_{11} & a_{12} + a'_{12} \\ ... & ... \\ \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix} = $ $\begin{vmatrix} (a_{11} + a'_{11}) b_{11} + (a_{12} + a'_{12}) b_{21} & (a_{11} + a'_{11}) b_{12} + (a_{12} + a'_{12}) b_{22}\\ ... & ... \\ \end{vmatrix} = $ $\begin{vmatrix} a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22}\\ ... & ... \\ \end{vmatrix} +$ $\begin{vmatrix} a'_{11} b_{11} + a'_{12} b_{21} & a'_{11} b_{12} + a'_{12} b_{22}\\ ... & ... \\ \end{vmatrix}$
thus $D(A)$ obeys the same properties as $\text{det } A$, so $D(A) = \text{det } A$ and we have $\text{det } AB = \text{det } A \cdot \text{det } B$
$I = A^{-1} A$
$\text{det }I = \text{det }A^{-1} A$
$\text{det }A^{-1} \cdot \text{det } A = 1$
$\text{det }A^{-1} = \cfrac{1}{\text{det } A}$
Consequence 2:
now can take into account the permutation matrix $P$ in the $PA = LU$ decomposition
$\text{det } PA = \text{det } LU$
$\text{det } P \cdot \text{det } A = \text{det } L \cdot \text{det } U$
$\text{det } P = \pm 1$, $\text{det } L = 1$ (elements on the diagonal of $L$ are 1's)
so $\text{det } A = \text{det } P \cdot \text{det } U$
Property 10: Transposition
$\text{det } A^T = \text{det } A$
The transpose of $A$ has the same determinant as $A$
consider $PA = LU$ factorization
transpose: $A^T P^T = U^T L^T$
take determinant, apply property 10 and compare $\text{det }P \cdot \text{det }A = \text{det } L \cdot \text{det } U$ with $\text{det }A^T \cdot \text{det }P^T = \text{det }U^T \cdot \text{det } L^T$
$\text{det } L = \text{det } L^T = 1$ (both have 1's on the diagonal)
$\text{det } U = \text{det } U^T = \prod d_i$ - they have the same elements on the diagonal
finally $\text{det } P = \text{det } P^T$ because $P^T P = I$ ($P$ is orthogonal) and by property 9 have $\text{det } P^T \cdot \text{det } P = 1$. That happens only when they agree on the sign.
thus, $\text{det } A^T = \text{det } A$
all the properties above are applied to rows, but the property #10 says that we can apply them to columns as well
Calculating Determinants
There are several possible ways to calculate determinants:
the Determinant Formula
the Pivot Formula
Cofactors
Determinant Formula
Let's try to find out how we can compute the determinant using the properties
$2 \times 2$ case
a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix} = \ ...$
can use the property 3 to divide the problem into smaller parts, and solve them separately
$... \ = \begin{vmatrix}
a_{11} + 0 & 0 + a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix} \ \mathop{=}\limits^{3^{\circ}} \ \begin{vmatrix} a_{11} & 0 \\ a_{21} & a_{22} \\ \end{vmatrix} + \begin{vmatrix} 0 & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix} = \underbrace{\begin{vmatrix}
a_{11} & 0 \\
\end{vmatrix}}_{0} +
\begin{vmatrix} a_{11} & 0 \\ 0 & a_{22} \\ \end{vmatrix} + \begin{vmatrix} 0 & a_{12} \\ a_{21} & 0 \\ \end{vmatrix} + \underbrace{\begin{vmatrix} 0 & a_{12} \\ 0 & a_{22} \\ \end{vmatrix}}_{0} = \ ...$
by property 6 (zero row) and 10 (determinant of transpose), we know that some parts are 0. so we're left with
a_{11} & 0 \\ 0 & a_{22} \\ \end{vmatrix} + \begin{vmatrix} 0 & a_{12} \\ a_{21} & 0 \\ \end{vmatrix} = \ ...$
now can change the rows of the second summand by property 2 (sign reversal) and get
a_{11} & 0 \\ 0 & a_{22} \\ \end{vmatrix} - \begin{vmatrix} a_{21} & 0 \\ 0 & a_{12} \\ \end{vmatrix} = a_{11}a_{22} - a_{21}a_{12}$
Can do the same for $3 \times 3$ matrices
a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{vmatrix} = \ ...$
we follow the same divide and conquer approach
most of the terms will go away because they will be equal to 0
the "survivers" will have one non-zero entry from each row
so for $3 \times 3$ we have:
a_{11} & 0 & 0\\ 0 & a_{22} & 0\\ 0 & 0 & a_{33}\\ \end{vmatrix} + \begin{vmatrix} a_{11} & 0 & 0\\ 0 & 0 & a_{23}\\ 0 & a_{32} & 0\\ \end{vmatrix} + \begin{vmatrix} 0 & a_{12} & 0\\ a_{21} & 0 & 0\\ 0 & 0 & a_{33}\\ \end{vmatrix} + \begin{vmatrix} 0 & a_{12} & 0\\ 0 & 0 & a_{23}\\ a_{31} & 0 & 0\\ \end{vmatrix} + \begin{vmatrix} 0 & 0 & a_{13}\\ a_{21} & 0 & 0\\ 0 & a_{32} & 0\\ \end{vmatrix} + \begin{vmatrix} 0 & 0 & a_{13}\\ 0 & a_{22} & 0\\ a_{31} & 0 & 0\\ \end{vmatrix}$
let's have a closer look at each of them
a_{11} & 0 & 0\\ 0 & a_{22} & 0\\ 0 & 0 & a_{33}\\ \end{vmatrix} = a_{11}a_{22}a_{33}$, diagonal and nice
a_{11} & 0 & 0\\ 0 & 0 & a_{23}\\ 0 & a_{32} & 0\\ \end{vmatrix} = - a_{11}a_{23}a_{32}$ - need 1 row exchange to transform it to $I$-like form
0 & a_{12} & 0\\ a_{21} & 0 & 0\\ 0 & 0 & a_{33}\\ \end{vmatrix} = - a_{12}a_{21}a_{33}$ - also 1 flip away
0 & a_{12} & 0\\ 0 & 0 & a_{23}\\ a_{31} & 0 & 0\\ \end{vmatrix} = a_{12}a_{23}a_{31}$ - 2 exchanges,
0 & 0 & a_{13}\\ a_{21} & 0 & 0\\ 0 & a_{32} & 0\\ \end{vmatrix} = a_{13}a_{21}a_{32}$ - 2 exchanges
0 & 0 & a_{13}\\ 0 & a_{22} & 0\\ a_{31} & 0 & 0\\ \end{vmatrix} = -a_{13}a_{22}a_{31}$ - 3 exchanges
So, the formula for $3 \times 3$:
a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{vmatrix} = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} - a_{13}a_{22}a_{31}$
or, schematically:
$n \times n$ case: "Big Formula"
The big formula:
we consider all $n!$ possible permutation matrices $P$
why $n!$? we can choose an element from the row 1 in $n$ ways, an element from the row 2 in $n - 1$ ways, ..., the last - in one way
$\text{det } A = \sum\limits_{\text{$n!$ permutations $P$}} \text{det } P \cdot a_{1\alpha_1} a_{2\alpha_2} ... a_{n\alpha_n}$
where $\boldsymbol \alpha = (\alpha_1, \ ... \ , \alpha_n)$ is a Permutation of $(1, \ ... \ , n)$
The easiest way is to use the properties 2, 5, 7 and 9:
do the factorization $PA = LU$
know that $\text{det } A = \text{det } P \cdot \text{det } U$
if one of the pivots is 0, then $\text{det } A = 0$
if $P$ is $\pm 1$, depending on the number of permutations, and $\text{det } U = \prod d_i$
Cofactors give a way to break $n \times n$ determinant to $(n - 1) \times (n - 1)$ determinants
$3 \times 3$ Case: Intuition
Suppose $A$ is a $3 \times 3$ matrix
let's group them
$\text{det } A = a_{11} (a_{22}a_{33} - a_{23}a_{32}) + a_{12} (-1) (a_{21}a_{33} - a_{23} a_{33}) + a_{13} (a_{21}a_{32} - a_{22}a_{31})$
note that now in parentheses we have determinants of smaller matrices!
for $a_{11}$ we have $a_{22}a_{33} - a_{23}a_{32} = \begin{vmatrix}
a_{22} & a_{23}\\ a_{32} & a_{33}\\ \end{vmatrix}$
for $a_{12}$ we have $- (a_{21}a_{33} - a_{23} a_{33}) = - \begin{vmatrix}
a_{21} & a_{23}\\ a_{31} & a_{33}\\ \end{vmatrix}$ (note the $-$ sign!)
these are co-factors of $a_{11}, a_{12}, a_{13}$ respectively
so we can write this as
a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{vmatrix} = \begin{vmatrix} a_{11} & 0 & 0\\ 0 & a_{22} & a_{23}\\ 0 & a_{32} & a_{33}\\ \end{vmatrix} - \begin{vmatrix} 0 & a_{12} & 0\\ a_{21} & 0 & a_{23}\\ a_{31} & 0 & a_{33}\\ \end{vmatrix} + \begin{vmatrix} 0 & 0 & a_{13}\\ a_{21} & a_{22} & 0\\ a_{31} & a_{32} & 0\\ \end{vmatrix}$
a cofactor of $a_{ij}$ is $C_{ij}$
$C_{ij}$ is a determinant of a $n - 1$ matrix - it's a matrix $A$ with row $i$ and column $j$ removed
note that we can have a minus sign before some of the cofactors
we have $C_{ij}$ with $-$ if $i+j$ is odd, and $+$ if $i+j$ is even
so for $3 \times 3$ matrix we take signs this way: $\begin{vmatrix}
+ & - & + \\ - & + & - \\ + & - & + \\ \end{vmatrix}$
a minor of $a_{ij}$ is $M_{ij}$
it's the same as cofactor, but always with the same sign
The Cofactor Formula
We can take co-factors along any row or column
suppose we take it along row 1
then the formula is $\text{det } A = a_{11} C_{11} + a_{12} C_{12} + \ ... \ + a_{1n} C_{1n}$
What can we do with determinants?
Cramer's Rule
through the Cramer's Rule:
Find the inverse and solve a System of Linear Equations
$\text{det } A$ = volume of a parallelepiped formed by vector-rows of $A$
$A = \begin{bmatrix}
a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{bmatrix}$
$\mathbf r_1 = \Big[a_{11} \ \ a_{12} \ \ a_{13} \Big]$
$| \text{det } A|$ is the volume of the box formed by vectors $\mathbf r_1, \mathbf r_2, \mathbf r_3$
To check if this is indeed true, we need to verify that the volume obeys the 3 defining properties
$A = I$ works, the volume is 1
property 2: reversing two rows changes the sign (don't care), but the volume remains the same - true
linearity:
3a. suppose we double one edge: the volume double
3b. see pictorially
Area of Triangle
We know how to compute the area of a square
so we can compute the area of a triangle!
let $A$ be $2 \times 2$ matrix, $A = \begin{bmatrix}
a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}$
the area of a triangle that starts in origin is $\cfrac{1}{2} \text{det } A = \cfrac{1}{2} (a_{11} \, a_{22} - a_{12} \, a_{21}) $
What is it doesn't start in origin?
then we calculate it by taking the following determinant:
x_1 & y_1 & 1 \\ x_1 & y_1 & 1 \\ x_1 & y_1 & 1 \\ \end{vmatrix}$
Linear Algebra MIT 18.06 (OCW)
Strang, G. Introduction to linear algebra.
Курош А.Г. Курс Высшей Алгебры
Retrieved from "http://mlwiki.org/index.php?title=Determinants&oldid=536"