In attempting to solve a linear system of equations for x¹, x² and x³

(1)

one is led in a very natural way to consider the square array

(2)

We have written a_ij = a_jⁱ,i,j = 1, 2, 3. The solution of (1) requires that we attach a numerical value to the matrix of elements (2). We do this in the following way: We attach 3³ = 27 numerical values to a set of 𝜖^ijk,i,j,k = 1, 2, 3. If at least two of the superscripts in 𝜖^ijk are the same, the value of 𝜖^ijk is zero. Thus 𝜖²²³ = 𝜖¹³¹ = 𝜖³³³ = 0, etc. If the i, j, k are all different, the value of 𝜖^ijk is to be +1 or −1 according to whether it takes an even or odd number of permutations to rearrange the ijk into the natural order 123. Let us lood at 𝜖³²¹ 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 𝜖¹²³ = −1. We have

We now define

(3)

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

Only 3! = 6 terms occur in the exapansion of (3) since there are 3! permutations of 123. All other values of 𝜖_ijk are zero.

We can define 𝜖_ijk in exactly the same manner in which the 𝜖^ijk were defined. We leave it to the reader to show that

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 nth order determinants is simple. We define the 𝜖^i₁i₂i_n to have the following numerical values: 𝜖^i₁i₂i_n = 0 if at least two of the superscripts are the same. The values of the superscripts range from 1 to n. If the i₁,i₂,…,i_n are distinct, the value of 𝜖^i₁i₂i_n is to be +1 or −1 depending on whether an even or odd number of permutations is required to rearrange i₁,i₂,…,i_n into the natural order 123n. The numerical value (determinant) of the square array of elements , i,j = 1, 2,…,n, is defined as

(4)

where the 𝜖_i1i2in are defined in precisely the same manner in which the 𝜖^i₁i₂i_n are defined. In general, (4) consists of n! terms, each term a product of elements, one element fom each row and column of .

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

(5)

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

(6)

But 𝜖^ijka _i³a _j²a _k¹ = 𝜖^ijka _k¹a _j²a _i¹ = 𝜖^kjia _i¹a _j²a _k³, 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^ijk is replaced by 𝜖^kji. Since 𝜖^ijk = −𝜖^kji, 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

(7)

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

(8)

since 𝜖^ijka _i³a _j²a _k³ = 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 scalar multiple of the corresponding elements of another row (or column).

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

1 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].