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Solution of linear systems by determinants. Properties of determinants. Evaluation of determinants. Minor. Cofactor. Cramer’s Rule. Homogeneous systems.

Def. Determinant. A square array of quantities, called elements, symbolizing the sum of certain products of these elements. The symbol

denotes a determinant of order n. It is an abbreviation for the algebraic sum of all possible products

where each product of n factors contains one and only one element from each row and one and
only one element from each column. There will be n! such products. Each product has a plus or
minus sign attached to it according as the column indices form an even or odd permutation when
the row indices are in natural order (i.e. 1, 2, 3, ... ). For example, the term a_{13}a_{21}a_{34}a_{42} of the
expansion of a determinant of order four has the column indices in order (3,1,4,2) . This term
should have a negative sign attached, since three successive interchanges will change the column
indices to (1,3,4,2), (1,3,2,4) and (1,2,3,4), the last being in natural order.

Determinants of the second order. The value of the determinant

of order two is given by

The solution of a system of two linear equations in two unknowns given in terms of determinants. The solution of the linear system

is

providing the equations are consistent and independent. The equations are consistent and independent if and only if

Note that the denominator determinants are the same for both x and y and consist of the
coefficients of the variables x and y arranged exactly as they appear in the left members of 1).
This denominator determinant is called the determinant of the coefficients or the determinant
of the system. The numerator determinants are constructed from this denominator determinant
by replacing the column containing the coefficients of the variable being solved for by the
column of constants, c_{1} and c_{2}, from the right side of 1) i.e. the numerator determinant in the
solution for x is constructed from the denominator determinant by replacing the coefficients a_{1}
and a_{2} of x by c_{1} and c_{2} and likewise in the solution for y.

Determinants of the third order. The value of the determinant

is given by

This sum can be remembered by the device shown in Fig. 1 where elements connected by red lines correspond to positive products and elements connected by blue lines represent negative products.

The solution of a system of three linear equations in three unknowns given in terms of determinants. The solution of the linear system

is given by

providing the equations are consistent and independent. The equations are consistent and independent if and only if

Determinants are most easily evaluated by a technique employing the following properties of determinants.

Properties of determinants.

1. If all the elements of a column (or row) are zero, the value of the determinant is zero.

2. If each of the elements in a row (or column) of a determinant is multiplied by the same number p, the value of the determinant is multiplied by p.

3. If two columns (or rows) are identical, the value of the determinant is zero.

4. Interchanging any two rows (or columns) reverses the sign of the determinant.

5. The value of a determinant is unaltered when all the corresponding rows and columns are interchanged. Thus any theorem proved true for rows holds for columns, and conversely.

Example.

6. If each element of a row (or column) of a determinant is expressed as the sum of two (or more) terms, the determinant can be expressed as the sum of two (or more) determinants.

Example.

7. If to each element of a row (or column) of a determinant is added m times the corresponding element of another row (or column) the value of the determinant is not changed.

Def. Minor of an element in a determinant. The determinant, of next lower order, obtained by striking out the row and column in which the element lies.

Def. Cofactor of an element in a determinant. Denote the minor of element a_{ij}
of the i-th row and j-th column of a determinant |A| by |M_{ij }|. The cofactor α_{ij} of the element a_{ij} is
given as the signed minor (-1)^{i+j} |M_{ij }| i.e. it is the minor, taken with a positive or negative sign,
according as the sum of the column number and the row number is even or odd.

Example. Let

Then the minor of element a_{21} is

The cofactor of element a_{21} is

Evaluation of a determinant by minors.

Theorem 1. The value of a determinant of order n is equal to the sum of the products formed by multiplying each element of any selected row (or column) by its cofactor. The value of a determinant of order 1 is the value of the single element of the determinant.

Example. Let

Then

Where α_{11}, α_{12}, and α_{13} are the cofactors of a_{11}, a_{11}, and a_{13.}

The usual procedure for evaluating a determinant of order greater than 3 is the following:

Step 1. For some selected row or column use property 7 above to reduce to zero all elements except one.

Step 2. Expand the elements of the selected row (or column) using Theorem 1. This yields a single determinant of lower order than the original determinant.

Step 3. Repeat Step 2 until the remaining determinant is of the second or third order.

Example. Evaluate the determinant

Solution. In general, the first step in the procedure is to locate an element in the matrix that is either 1 or -1. If one doesn’t exist, we create one using property 2 or 7 listed above. In this case there is a -1 in the second column. We will use it to reduce to zero the other elements in the same column. Adding the third row to the first gives

0 0 -2 7

for the new first row.

Adding twice the third row to the second gives

-1 0 4 7

for the new second row.

Subtracting three times the third row from the fourth gives

10 0 -1 -8

for the new fourth row.

We thus have

We now expand the determinant along the elements of the second column to obtain

The solution of a system of n linear equations in n unknowns by determinants.

Cramer’s Rule. Given a system of n linear equations in n unknowns

where the n unknowns in the left members are assumed to appear in the same order in all
equations and the constant terms, k_{1}, k_{2}, ... , k_{n}, are in the right members. Let D denote the
determinant of the coefficients formed from the coefficients in the left members arranged just
as they appear in the system. Let N_{x} be the determinant formed from D by replacing the
coefficients of x with the constant terms k_{1}, k_{2}, ... , k_{n}, let N_{y} be the determinant formed from D
by replacing the coefficients of y with the constant terms k_{1}, k_{2}, ... , k_{n}, etc. Then:

Case 1. D ≠ 0. The system has a unique solution given by

x = N_{x }/D, y = N_{y }/D, z = N_{z }/D, ............

Case 2. D = 0 and at least one of N_{x }, N_{y }, N_{z} , ... ≠ 0. The system has no solution. It
is inconsistent.

Case 3. D = 0 and N_{x }, N_{y }, N_{z} , ... = 0. The system may or may not have a solution. If it
has a solution, the equations are dependent and there will be an infinite number of solutions. If it
doesn’t have a solution, the equations are inconsistent.

Linear systems containing fewer equations than unknowns. Ordinarily if there are fewer equations than unknowns in a system, the system will have an infinite number of solutions. However, if the system contains inconsistent equations, there will be no solution.

To solve a consistent system of m equations in n unknowns, where m < n, solve for m of the unknowns in terms of the others.

Linear systems containing more equations than unknowns. Ordinarily if there are more equations than unknowns, the system is inconsistent. However, this need not be the case and the system may contain a number of dependent equations; in which case there may be either one or an infinite number of solutions.

Homogeneous systems. A linear equation is homogeneous if its constant term is zero. Every system of homogeneous equations

has the so-called trivial solution x = 0, y = 0, z = 0, .....

Theorem. A necessary and sufficient condition that a system of n homogeneous linear equations in n unknowns have solutions other than the trivial solution is that its determinant of the coefficients is zero. If the determinant of the coefficients is zero, it will have an infinite number of solutions.

References

Hawks, Luby, Touton. Second-Year Algebra

Murray R. Spiegel. College Algebra (Schaum)

Raymond W. Brink. A First Year of College Mathematics

Frank Ayres. First Year College Mathematics (Schaum)

James / James. Mathematics Dictionary

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