Comprehensive Study Notes

Section 8.5 discusses a kind of variation function that gives rise to an equation involving a determinant. Therefore, we now discuss determinants.

A determinant is a square array of $n^{2}$ quantities (called elements); the value of the determinant is calculated from its elements in a manner to be given shortly. The number $n$ is the order of the determinant. Using $a_{i j}$ to represent a typical element, we write the $n$ th-order determinant as

\(
\operatorname{det}\left(a{i j}\right)=\left|\begin{array}{ccccc}
a{11} & a{12} & a{13} & \cdots & a{1 n} \tag{8.20}\
a{21} & a{22} & a{23} & \cdots & a{2 n} \
\cdot & \cdot & \cdot & \cdots & \cdot \
\cdot & \cdot & \cdot & \cdots & \cdot \
\cdot & \cdot & \cdot & \cdots & \cdot \
a{n 1} & a{n 2} & a{n 3} & \cdots & a_{n n}
\end{array}\right|
\)

The vertical lines in (8.20) have nothing to do with absolute value. Before considering how the value of the $n$ th-order determinant is defined, we consider determinants of first, second, and third orders.

A first-order determinant has one element, and its value is simply the value of that element. Thus

\(
\begin{equation}
\left|a{11}\right|=a{11} \tag{8.21}
\end{equation}
\)

where the vertical lines indicate a determinant and not an absolute value.
A second-order determinant has four elements, and its value is defined by

\(
\left|\begin{array}{ll}
a{11} & a{12} \tag{8.22}\
a{21} & a{22}
\end{array}\right|=a{11} a{22}-a{12} a{21}
\)

The value of a third-order determinant is defined by

\(
\left|\begin{array}{lll}
a{11} & a{12} & a{13} \tag{8.23}\
a{21} & a{22} & a{23} \
a{31} & a{32} & a{33}
\end{array}\right|=a{11}\left|\begin{array}{ll}
a{22} & a{23} \
a{32} & a{33}
\end{array}\right|-a{12}\left|\begin{array}{ll}
a{21} & a{23} \
a{31} & a{33}
\end{array}\right|+a{13}\left|\begin{array}{ll}
a{21} & a{22} \
a{31} & a{32}
\end{array}\right|
\)

\(
\begin{align}
= & a{11} a{22} a{33}-a{11} a{32} a{23}-a{12} a{21} a{33}+a{12} a{31} a{23} \
& +a{13} a{21} a{32}-a{13} a{31} a{22} \tag{8.24}
\end{align}
\)

A third-order determinant is evaluated by writing down the elements of the top row with alternating plus and minus signs and then multiplying each element by a certain secondorder determinant; the second-order determinant that multiplies a given element is found by crossing out the row and column of the third-order determinant in which that element appears. The $(n-1)$-order determinant obtained by striking out the $i$ th row and the $j$ th column of the $n$ th-order determinant is called the minor of the element $a{i j}$. We define the cofactor of $a{i j}$ as the minor of $a{i j}$ times the factor $(-1)^{i+j}$. Thus (8.23) states that a thirdorder determinant is evaluated by multiplying each element of the top row by its cofactor and adding up the three products. [Note that (8.22) conforms to this evaluation by means of cofactors, since the cofactor of $a{11}$ in (8.22) is $a{22}$, and the cofactor of $a{12}$ is $-a_{21}$.] A numerical example is

\(
\begin{aligned}
\left|\begin{array}{rrr}
5 & 10 & 2 \
0.1 & 3 & 1 \
0 & 4 & 4
\end{array}\right| & =5\left|\begin{array}{ll}
3 & 1 \
4 & 4
\end{array}\right|-10\left|\begin{array}{rr}
0.1 & 1 \
0 & 4
\end{array}\right|+2\left|\begin{array}{rr}
0.1 & 3 \
0 & 4
\end{array}\right| \
& =5(8)-10(0.4)+2(0.4)=36.8
\end{aligned}
\)

Denoting the minor of $a{i j}$ by $M{i j}$ and the cofactor of $a{i j}$ by $C{i j}$, we have

\(
\begin{equation}
C{i j}=(-1)^{i+j} M{i j} \tag{8.25}
\end{equation}
\)

The expansion (8.23) of the third-order determinant can be written as

\(
\operatorname{det}\left(a{i j}\right)=\left|\begin{array}{lll}
a{11} & a{12} & a{13} \tag{8.26}\
a{21} & a{22} & a{23} \
a{31} & a{32} & a{33}
\end{array}\right|=a{11} C{11}+a{12} C{12}+a{13} C{13}
\)

A third-order determinant can be expanded using the elements of any row and the corresponding cofactors. For example, using the second row to expand the third-order determinant, we have

\(
\begin{gather}
\operatorname{det}\left(a{i j}\right)=a{21} C{21}+a{22} C{22}+a{23} C{23} \tag{8.27}\
\operatorname{det}\left(a{i j}\right)=-a{21}\left|\begin{array}{ll}
a{12} & a{13} \
a{32} & a{33}
\end{array}\right|+a{22}\left|\begin{array}{ll}
a{11} & a{13} \
a{31} & a{33}
\end{array}\right|-a{23}\left|\begin{array}{ll}
a{11} & a{12} \
a{31} & a_{32}
\end{array}\right| \tag{8.28}
\end{gather}
\)

and expansion of the second-order determinants shows that (8.28) is equal to (8.24). We may also use the elements of any column and the corresponding cofactors to expand the determinant, as can be readily verified. Thus for the third-order determinant, we can write

\(
\begin{array}{ll}
\operatorname{det}\left(a{i j}\right)=a{k 1} C{k 1}+a{k 2} C{k 2}+a{k 3} C{k 3}=\sum{l=1}^{3} a{k l} C{k l}, & k=1 \text { or } 2 \text { or } 3 \
\operatorname{det}\left(a{i j}\right)=a{1 k} C{1 k}+a{2 k} C{2 k}+a{3 k} C{3 k}=\sum{l=1}^{3} a{l k} C{l k}, & k=1 \text { or } 2 \text { or } 3
\end{array}
\)

The first expansion uses one of the rows; the second uses one of the columns.

We define determinants of higher order by an analogous row (or column) expansion. For an $n$ th-order determinant,

\(
\begin{equation}
\operatorname{det}\left(a{i j}\right)=\sum{l=1}^{n} a{k l} C{k l}=\sum{l=1}^{n} a{l k} C_{l k}, \quad k=1 \text { or } 2 \text { or } \ldots \text { or } n \tag{8.29}
\end{equation}
\)

Some theorems on determinants are as follows (for proofs, see Sokolnikoff and Redheffer, pp. 702-707):
I. If every element of a row (or column) of a determinant is zero, the value of the determinant is zero.
II. Interchanging any two rows (or columns) multiplies the value of a determinant by -1 .
III. If any two rows (or columns) of a determinant are identical, the determinant has the value zero.
IV. Multiplication of each element of any one row (or any one column) by some constant $k$ multiplies the value of the determinant by $k$.
V. Addition to each element of one row of the same constant multiple of the corresponding element of another row leaves the value of the determinant unchanged. This theorem also applies to the addition of a multiple of one column to another column.
VI. The interchange of all corresponding rows and columns leaves the value of the determinant unchanged. (This interchange means that column one becomes row one, column two becomes row two, etc.)

EXAMPLE

Use Theorem V to evaluate

\(
B=\left|\begin{array}{llll}
1 & 2 & 3 & 4 \tag{8.30}\
4 & 1 & 2 & 3 \
3 & 4 & 1 & 2 \
2 & 3 & 4 & 1
\end{array}\right|
\)

Addition of -2 times the elements of row one to the corresponding elements of row four changes row four to $2+(-2) 1=0,3+(-2)(2)=-1,4+(-2) 3=-2$, and $1+(-2) 4=-7$. Then, addition of -3 times row one to row three and -4 times row one to row two gives

\(
B=\left|\begin{array}{rrrr}
1 & 2 & 3 & 4 \tag{8.31}\
0 & -7 & -10 & -13 \
0 & -2 & -8 & -10 \
0 & -1 & -2 & -7
\end{array}\right|=1\left|\begin{array}{rrr}
-7 & -10 & -13 \
-2 & -8 & -10 \
-1 & -2 & -7
\end{array}\right|
\)

where we expanded $B$ in terms of elements of the first column. Subtracting twice row three from row two and seven times row three from row one, we have

\(
B=\left|\begin{array}{rrr}
0 & 4 & 36 \tag{8.32}\
0 & -4 & 4 \
-1 & -2 & -7
\end{array}\right|=(-1)\left|\begin{array}{rr}
4 & 36 \
-4 & 4
\end{array}\right|=-(16+144)=-160
\)

The diagonal of a determinant that runs from the top left to the lower right is the principal diagonal. A diagonal determinant is a determinant all of whose elements are zero except those on the principal diagonal. For a diagonal determinant,

\(
\begin{align}
\left|\begin{array}{ccccc}
a{11} & 0 & 0 & \cdots & 0 \
0 & a{22} & 0 & \cdots & 0 \
0 & 0 & a{33} & \cdots & 0 \
\cdot & \cdot & \cdot & \cdots & \cdot \
0 & 0 & 0 & \cdots & a{n n}
\end{array}\right| & =a{11}\left|\begin{array}{cccc}
a{22} & 0 & \cdots & 0 \
0 & a{33} & \cdots & 0 \
\cdot & \cdot & \cdots & \cdot \
0 & 0 & \cdots & a{n n}
\end{array}\right|=a{11} a{22}\left|\begin{array}{cccc}
a{33} & 0 & \cdots & 0 \
0 & a{44} & \cdots & 0 \
\cdot & \cdot & \cdots & \cdot \
0 & 0 & \cdots & a{n n}
\end{array}\right| \
& =\cdots=a{11} a{22} a{33} \ldots a_{n n} \tag{8.33}
\end{align}
\)

A diagonal determinant is equal to the product of its diagonal elements.
A determinant whose only nonzero elements occur in square blocks centered about the principal diagonal is in block-diagonal form. If we regard each square block as a determinant, then a block-diagonal determinant is equal to the product of the blocks. For example,

The dashed lines outline the blocks. Equation (8.34) is readily proved by expanding the left side in terms of elements of the top row and expanding several subsequent determinants using their top rows (Prob. 8.21).