DETERMINANT AS SUM OF PRODUCTS OF ELEMENTS OF MATRICES

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DETERMINANT AS SUM OF PRODUCTS OF ELEMENTS

DETERMINANT AS SUM OF PRODUCTS OF ELEMENTS: Let [katex]A=\left[a_{ij}\right][/katex] be a square matrix of order n with entries from a field F.

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Then the sum

\sum_\sigma(-1)^k a_{1 k_1} a_{2 k_2} \cdots a_{n k_n}

where k is 0 or 1 according as the permutation

\sigma=\left(\begin{array}{ccccc}
1 & 2 & 3 & \cdots & n \\
k_1 & k_2 & k_3 & \cdots & k_n
\end{array}\right)

in [katex]S_n[/katex] is even or odd, is called the determinant of A and is denoted by det A or by

\left|\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n} \\
a_{21} & a_{22} & \cdots & a_{2 n} \\
\vdots & \vdots & \cdots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n}
\end{array}\right|.

In particular if n=3, then [katex]S_3[/katex] has 3!=6 elements which are

I=\left(\begin{array}{lll}1 & 2 & 3 \\ 1 & 2 & 3\end{array}\right),
\sigma_1=\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1\end{array}\right),
\sigma_2=\left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 2\end{array}\right),
\sigma_3=\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 3\end{array}\right),
\sigma_4=\left(\begin{array}{lll}1 & 2 & 3 \\ 1 & 3 & 2\end{array}\right),
\sigma_5=\left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 2 & 1\end{array}\right)

Of the above six permutations, the first three are even and the next three are old. So if

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] }

then

\text{det A}= \sum_\sigma(-1)^k a_{11} a_{2_2} a_{33} \\

= 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}
\text { Here } \quad \sigma=\left(\begin{array}{ccc}
1 & 2 & 3 \\
i_1 & i_2 & i_3
\end{array}\right)

is one of the permutations given above. Thus, for writing the products, we first take the subseript i in [katex]a_{ij}[/katex] as in the first row of the corresponding permutation and j as tha corresponding element below i in the second row of the permutation. This process is repeated for all the six permutations.

For example, the term in sum (2) corresponding to the first permutation namely

I=\left(\begin{array}{lll}
1 & 2 & 3 \\
1 & 2 & 3
\end{array}\right)

is the product [katex]a_{11} a_{22} a_{13}[/katex]. Likewise, for the permutation

\sigma_3=\left(\begin{array}{lll}
1 & 2 & 3 \\
2 & 1 & 3
\end{array}\right)

we have the product [katex]a_{12} a_{21} a_{33}[/katex] with minus sign because [katex]\sigma_3[/katex] is an odd permutation. Similarly, if

A=\left[\begin{array}{lll}
1 & 6 & 7 \\
2 & 3 & 5 \\
4 & 1 & 7
\end{array}\right] \text {, }

then, writing the corresponding values of [katex]a_{ij}[/katex] in the matrix A, we have:

\operatorname{det} A =1 \times 3 \times 7+6 \times 5 \times 4+7 \times 2 \times 1\\-6 \times 2 \times 7-1 \times 5 \times 1-7 \times 3 \times 4 \\
=21+120+14-84-5-84 \\
=155-173=-18

DETERMINANT AS SUM OF PRODUCTS OF ELEMENTS

For example let

A=\left[\begin{array}{rrr}
1 & -1 & 2 \\
3 & 1 & 4 \\
0 & -2 & 5
\end{array}\right],

then

 A^T=\left[\begin{array}{rrr}
1 & 3 & 0 \\
-1 & 1 & -2 \\
2 & 4 & 5
\end{array}\right]

and

\operatorname{det} A=16=\operatorname{det} A^T

Remark. The expansion of det A as given in Definition is called the expansion of determinant by rows or row expansion of det A. Similarly, one has the notion of column expansion of det A. Theorem shows that these two expansions are algebraically identical. Theorem. Let A be an [katex]n \times n[/katex] matrix and let det A denote the determinam function. Then (i) If the row vector [katex]a_i^r=\left[\begin{array}{llll}a_{i1} & a_{i 2} & \cdots & a_{\text {in }}\end{array}\right][/katex] of A is of the form

a_i^T=a b_i^T

then

det \,\ A=\alpha \,\  det \,\ B

where B is the matrix obtrinted from A by replacing its ith row [katex]a_i^T[/katex] by [katex]b_i^{\top}[/katex]. (ii) If a row [katex]a_i^T[/katex] of A is [katex]\theta[/katex], then

\operatorname{det} A=0

(iii) If a row [katex]a_i^T[/katex] of A is of the form

a_i^T=b_i^T+c_i^T

then where B and C are matrices obtained from A by replacing its ith row [katex]b_i^T[/katex] and [katex]c_i^T[/katex] respectively.

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