DETERMINANT OF A SQUARE MATRIX
The reader may be fimiliar with the concept of determinants. This concept played a basic role in the solution of systems of linear equations before the use of matrices and computers. The use of determinants is becoming less and less with every passing day. Neverthless determinants still play an important but minor role in finding solutions of linear problems.
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\begin{aligned}
&a_{11} x_1+a_{12} x_2=b_1 \\
&a_{21} x_1+a_{22} x_2=b_2
\end{aligned}in two unknowns [katex]x_1, x_2[/katex]. Here
A=\left[\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right], \quad A_b=\left[\begin{array}{ll|l}
a_{11} & a_{12} & b_1 \\
a_{21} & a_{22} & b_2
\end{array}\right]are the coefficient matrix and the augmented matrix respectively with entries as real numbers. By the Gauss-Jordan procedure, the solution of these equations is
x_1=\frac{\left(b_1 a_{22}-b_2 a_{12}\right)}{\left(a_{11} a_{22}-a_{21} a_{12}\right)}, x_2=\frac{\left(b_2 a_{11}-b_1 a_{21}\right)}{\left(a_{11} a_{22}-a_{21} a_{12}\right)}provided that
a_{11} a_{22}-a_{21} a_{12} \neq 0.The scalar [katex]a_{11} a_{22}-a_{21} a_{12}[/katex] is uniquely determined by the matrix A. It is called the determinant of order 2 of the square matrix A and is denoted by det A or |A|. Thus
|A|=\operatorname{det} A=a_{11} a_{22}-a_{21} a_{12}=\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right|.Note the two vertical bars instead of square brackets used for matrices. We note that det A of order 2 is a real number associated with a matrix A of order 2 So, in this case, we may regard det as a function whose domain is the set of all square matrices of order 2 and whose range is a subset of real numbers.
Propertics of determinants of order 2
The following propertics of determinants of order 2 follow directly from the above definition. (i) For any real number k,
\left|\begin{array}{ll}
k a_{11} & a_{12} \\
k a_{21} & a_{22}
\end{array}\right|=k\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right|(ii) If [katex]a_{12}=b_{12}+c_{12}, a_{22}=b_{22}+c_{22}[/katex], then
\begin{aligned}
\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right| &=\left|\begin{array}{ll}
a_{11} & b_{12}+c_{12} \\
a_{21} & b_{22}+c_{22}
\end{array}\right| \\
&=\left|\begin{array}{ll}
a_{11} & b_{12} \\
a_{21} & b_{22}
\end{array}\right|+\left|\begin{array}{ll}
a_{11} & c_{12} \\
a_{21} & c_{22}
\end{array}\right|.
\end{aligned}(iii) If the two columns of A are identical then det A=0. That is
\left|\begin{array}{ll}
a_{11} & a_{11} \\
a_{21} & a_{21}
\end{array}\right|=0 .(iv) The determinant of the unit matrix is 1 . That is
\left|\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right|=1 \text {. }It will be seen later that these properties are characteristics of a determinant of a square matrix of any order.
Determinant of Order n
Firstly we define the dresminant of an [katex]n \times n[/katex] matrix inductively. That is, from our knowledge of a detee wenant of order 2 , we define a determinant of order 3 and use this definition of a determinant of order 3 to describe a determinant of order 4 and so on. Thus a determinant of order 3 is defined as follows: Let
\begin{aligned}
&A=\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right] \\
&\text { Then } \operatorname{det} A=|A|=a_{11}\left|\begin{array}{ll}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{array}\right|-a_{12}\left|\begin{array}{cc}
a_{21} & a_{23} \\
a_{31} & a_{33}
\end{array}\right|+a_{13}\left|\begin{array}{cc}
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{array}\right|
\end{aligned}Note the minus sign before the second term on the right hand side of (1). For example, if
\begin{aligned}
A=\left[\begin{array}{lll}
4 & 1 & 2 \\
3 & 2 & 5 \\
1 & 2 & 3
\end{array}\right] \text {, then } \\
\operatorname{det} A=\left[\begin{array}{lll}
4 & 1 & 2 \\
3 & 2 & 5 \\
1 & 2 & 3
\end{array} \mid\right.&=4\left|\begin{array}{ll}
2 & 5 \\
2 & 3
\end{array}\right|-1\left|\begin{array}{ll}
3 & 5 \\
1 & 3
\end{array}\right|+2\left|\begin{array}{ll}
3 & 2 \\
1 & 2
\end{array}\right| \\
&=4(6-10)-1(9-5)+2(6-2) \\
&=-16-4+8=-12 .
\end{aligned}There is a simplex method to calculate a deterninant of order 3 . [katex]\begin{aligned} \operatorname{det} A &=a_{11}\left(a_{22} a_{13}-a_{23} a_{32}\right)-a_{12}\left(a_{11} a_{33}-a_{23} a_{31}\right)+a_{13}\left(a_{21} a_{32}-a_{22} a_{11}\right) \\ &=a_{11} a_{12} a_{13}+a_{12} a_{23} a_{31}+a_{13} a_{21} a_{12}-a_{11} a_{22} a_{12}-a_{12} a_{21} a_{33}-a_{13} a_{22} a_{31} \end{aligned}[/katex]
Sarrus’s Rule
We write the columns of A and adjoin to these the first two columns as below:
Now calculate the six products of numbers on the directed lines taking plus sign with those products on arrows pointing downwards and minus sign with products on arrows pointing upwards. Adding these products we get the value of det A. This is known as Sarrus’s rule. Note: The method of arrows given above works only for n=2, 3, and not for [katex]n \geq 4[/katex].