PlanetPhysics/Determinant

In attempting to solve a linear [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] of equations for $x^{1}$ , $x^{2}$ and $x^{3}$

$\begin{matrix} a_{1} x^{1} + b_{1} x^{2} + c_{1} x^{3} & = & d_{1} \\ a_{2} x^{1} + b_{2} x^{2} + c_{2} x^{3} & = & d_{2} \\ a_{3} x^{1} + b_{3} x^{2} + c_{3} x^{3} & = & d_{3} \end{matrix}$

one is led in a very natural way to consider the [[../PiecewiseLinear/|square]] array

$| \begin{matrix} a_{1}^{1} & a_{2}^{1} & a_{3}^{1} \\ a_{1}^{2} & a_{2}^{2} & a_{3}^{2} \\ a_{1}^{3} & a_{2}^{3} & a_{3}^{3} \end{matrix} |$

We have written $a_{i j} = a_{j}^{i}, i, j = 1, 2, 3$ . The solution of (1) requires that we attach a numerical value to the [[../Matrix/|matrix]] of elements (2). We do this in the following way: We attach $3^{3} = 27$ numerical values to a set of $ϵ^{i j k}, i, j, k = 1, 2, 3$ . If at least two of the superscripts in $ϵ^{i j k}$ are the same, the value of $ϵ^{i j k}$ is zero. Thus $ϵ^{223} = ϵ^{131} = ϵ^{333} = 0$ , etc. If the $i$ , $j$ , $k$ are all different, the value of $ϵ^{i j k}$ is to be $+ 1$ or $- 1$ according to whether it takes an even or odd number of permutations to rearrange the $i j k$ into the natural order $123$ . Let us lood at $ϵ^{321}$ and hence at the arrangement $321$ . Permuting the integers $2$ and $3$ permutes $321$ into $231$ , then permuting $3$ and $1$ permutes $231$ into $213$ , and finally $213$ permutes into $123$ if we interchange the integers $2$ and $1$ . Three (an odd number) permutations were required to permute $321$ into $123$ . Thus $ϵ^{123} = - 1$ . We have

$ϵ^{123} = ϵ^{312} = ϵ^{231} = + 1$ $ϵ^{213} = ϵ^{321} = ϵ^{132} = - 1$

We now define

$| \begin{matrix} a_{1}^{1} & a_{2}^{1} & a_{3}^{1} \\ a_{1}^{2} & a_{2}^{2} & a_{3}^{2} \\ a_{1}^{3} & a_{2}^{3} & a_{3}^{3} \end{matrix} | \equiv ϵ^{i j k} a_{i}^{1} a_{j}^{2} a_{k}^{3}$

The letters $i$ , $j$ , $k$ are indices of summation. Equation (3) defines the determinant of the [[../Matrix/|square matrix]] of elements (2) Its numerical value is given by the right-hand side of (3). It consists, in general, of $3! = 3 \cdot 2 \cdot 1 = 6$ terms, each term a product of three elements, one element from each row and column of (3). Expand (3) to get

$ϵ^{i j k} a_{i}^{1} a_{j}^{2} a_{k}^{3} = (a_{1}^{1} a_{2}^{2} a_{3}^{3} + a_{3}^{1} a_{1}^{2} a_{2}^{3} + a_{2}^{1} a_{3}^{2} a_{1}^{3}) - (a_{2}^{1} a_{1}^{2} a_{3}^{3} + a_{3}^{1} a_{2}^{2} a_{1}^{3} + a_{1}^{1} a_{3}^{2} a_{2}^{3})$

Only $3! = 6$ terms occur in the exapansion of (3) since there are $3!$ permutations of $123$ . All other values of $ϵ_{i j k}$ are zero.

We can define $ϵ_{i j k}$ in exactly the same manner in which the $ϵ^{i j k}$ were defined. We leave it to the reader to show that

$ϵ^{i j k} a_{i}^{1} a_{j}^{2} a_{k}^{3} = ϵ_{i j k} a_{1}^{i} a_{2}^{j} a_{3}^{k}$

The generalization of second and third order determinants (the order of a determinant is the number of rows or columns of the determinant) to the $n$ th order determinants is simple. We define the $ϵ^{i_{1} i_{2} \dots i_{n}}$ to have the following numerical values: $ϵ^{i_{1} i_{2} \dots i_{n}} = 0$ if at least two of the superscripts are the same. The values of the superscripts range from $1$ to $n$ . If the $i_{1}, i_{2}, \dots, i_{n}$ are distinct, the value of $ϵ^{i_{1} i_{2} \dots i_{n}}$ is to be $+ 1$ or $- 1$ depending on whether an even or odd number of permutations is required to rearrange $i_{1}, i_{2}, \dots, i_{n}$ into the natural order $123 \dots n$ . The numerical value (determinant) of the square array of elements $‖ a_{j}^{i} ‖$ , $i, j = 1, 2, \dots, n$ , is defined as

$| a_{j}^{i} | = | \begin{matrix} a_{1}^{1} & a_{2}^{1} & \dots & a_{n}^{1} \\ a_{1}^{2} & a_{2}^{2} & \dots & a_{n}^{2} \\ \dots & \dots & \dots & \dots \\ a_{1}^{n} & a_{2}^{n} & \dots & a_{n}^{n} \end{matrix} |$

$| a_{j}^{i} | = ϵ^{i_{1} i_{2} \dots i_{n}} a_{i_{1}}^{1} a_{i_{2}}^{2} \dots a_{i_{n}}^{n}$

$| a_{j}^{i} | = ϵ_{i_{1} i_{2} \dots i_{n}} a_{1}^{i_{1}} a_{2}^{i_{2}} \dots a_{n}^{i_{n}}$

where the $ϵ_{i_{1} i_{2} \dots i_{n}}$ are defined in precisely the same manner in which the $ϵ^{i_{1} i_{2} \dots i_{n}}$ are defined. In general, (4) consists of $n!$ terms, each term a product of elements, one element fom each row and column of $| a_{j}^{i} |$ .

To facilitate writing, we shall deal with third order determinants, but it will be obvious to the reader that any [[../Formula/|theorem]] derived for third order determinants will apply to determinants of any finite order. Le us consider

$Δ = | \begin{matrix} a_{1}^{1} & a_{2}^{1} & a_{3}^{1} \\ a_{1}^{2} & a_{2}^{2} & a_{3}^{2} \\ a_{1}^{3} & a_{2}^{3} & a_{3}^{3} \end{matrix} | = ϵ^{i j k} a_{i}^{1} a_{j}^{2} a_{k}^{3} = ϵ_{i j k} a_{1}^{i} a_{2}^{j} a_{3}^{k}$

We can obtain a new third order determinant by interchanging the first and third row of $Δ$ . This yields

$Δ^{'} = | \begin{matrix} a_{1}^{3} & a_{2}^{3} & a_{3}^{3} \\ a_{1}^{2} & a_{2}^{2} & a_{3}^{2} \\ a_{1}^{1} & a_{2}^{1} & a_{3}^{1} \end{matrix} | = ϵ^{i j k} a_{i}^{3} a_{j}^{2} a_{k}^{1}$

But $ϵ^{i j k} a_{i}^{3} a_{j}^{2} a_{k}^{1} = ϵ^{i j k} a_{k}^{1} a_{j}^{2} a_{i}^{1} = ϵ^{k j i} a_{i}^{1} a_{j}^{2} a_{k}^{3}$ , since $i$ amd $k$ are dummy indices. We see that every term of (6) is the same as every term in (5) with the exception that $ϵ^{i j k}$ is replaced by $ϵ^{k j i}$ . Since $ϵ^{i j k} = - ϵ^{k j i}$ , we conclude that $Δ = - Δ^{'}$ . we thus obtain the following theorems:

THEOREM 1.1 . Interchanging two rows (or columns) of a determinant changes the sign of the determinant.

THEOREM 1.1 . If two rows (or columns) of a determinant are the same, the value of the determinant is zero.

We note that

$Δ^{''} = | \begin{matrix} l a_{1}^{1} & l a_{2}^{1} & l a_{3}^{1} \\ a_{1}^{2} & a_{2}^{2} & a_{3}^{2} \\ a_{1}^{3} & a_{2}^{3} & a_{3}^{3} \end{matrix} | = ϵ^{i j k} (l a_{i}^{1}) a_{j}^{2} a_{k}^{3} = l Δ$

THEOREM 1.3 . If a row (or column) of a determinant is multiplied by a factor $l$ , the value of the determinant is thereby multiplied by $l$ .

Let us now investigate the determinant

$Δ^{'''} = | \begin{matrix} a_{1}^{1} + l a_{1}^{3} & a_{2}^{1} + l a_{2}^{3} & a_{3}^{l} + l a_{3}^{3} \\ a_{1}^{2} & a_{2}^{2} & a_{3}^{2} \\ a_{1}^{3} & a_{2}^{3} & a_{3}^{3} \end{matrix} |$

$Δ^{'''} = ϵ^{i j k} (a_{i}^{1} + l a_{i}^{3}) a_{j}^{2} a_{k}^{3}$

$Δ^{'''} = ϵ^{i j k} a_{i}^{1} a_{j}^{2} a_{k}^{3} + l ϵ^{i j k} a_{i}^{3} a_{j}^{2} a_{k}^{3}$

$Δ^{'''} = Δ$

since $ϵ^{i j k} a_{i}^{3} a_{j}^{2} a_{k}^{3} = 0$ from Theorem (1.2). Hence we obtain the following theorem:

THEOREM 1.4 . The value of a determinant remains unchanged if to the elements of any row (or column) is added a [[../Vectors/|scalar]] multiple of the corresponding elements of another row (or column).

The theorems derived above are very useful in evaluating a determinant.

References

[1] Lass, Harry. "Elements of pure and applied mathematics" New York: McGraw-Hill Companies, 1957.

This entry is a derivative of the Public domain work [1].

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PlanetPhysics/Determinant

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