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In eq. (40), the right-hand side is the result of adding, to each element of the rst column, the corresponding element of the third column multiplied by m.* We can show that the two sides of eq. (40) are equal as follows. First apply, to the right-hand side, property 6 and then property 5; if we do this, the right-hand side becomes      a11 a12 a13   a13 a12 a13           a21 a22 a23  m a23 a22 a23       a31 a32 a33   a33 a32 a33  Notice, now, that the second determinant above has two identical columns, and is thus equal to zero by property 4, proving that the two sides of eq. (40) are equal. Property 7 can sometimes be used to greatly reduce the amount of work needed to nd the value of a given determinant. This is accomplished by using property 7 to transform a given determinant into an equivalent determinant having more zeros as elements than the given determinant has. The following example will illustrate the procedure. Example
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Given the determinant  5 1   D 3 2   2 4  2    1  3
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(a) Find the value of D by expanding, in the usual way, the determinant in terms of, say, the elements of the second column.
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Solution
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    5 2  2    4 3 1 3 answer:
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9 2 2 15 4 4 5 6 7 38 44 75;
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* Property 7 applies to any combination of two columns, or two rows, we might wish to use.
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CHAPTER 3 Determinants and Equations
(b) Going back to the given determinant, nd D by making use of property 7. Solution One way is as follows. To each element of row 2 add the corresponding element of row 1 multiplied by 2. Next, to each element of row 3 add the corresponding element of row 1 multiplied by 4; thus,   5   D  3 10   2 20 1 2 2 4 4   2   5     1 4   7   3 8   22 1 0 0  2    5  5 
Notice now that all the elements in column 2 except one are zeros; this makes it easy to expand the determinant in terms of the elements of column 2, thus,    7 5  35 110 75; answer; as before:  D  22 5  Problem 35 Given that
  6   D  24   12
 5   16 120   1 25  3
use property 5 as an aid in nding the value of D. Problem 36 Find the value of
  3   1    8   2
6 0 0 0
10 5 20 15
 7  14    14   14 
Problem 37 In a determinant, if the elements of any given row (or column) are added to, or subtracted from, the corresponding elements of any other row (or column), is the value of the determinant changed (The given row or column remains, of course, unchanged in its original position.) Problem 38 By inspection, verify that
 3  4   1  6
1 2 3 0 4 4 7 5
 6  8   0 2  12 
Problem 39 Verify that
CHAPTER 3 Determinants and Equations
 0  2   0 1   3  1   3 2   2 4  0  4   1  596  1    0
1 0 2
5 2 0
0 1 3 2
Determinant Solution of Linear Simultaneous Equations
A linear or rst degree equation is one in which the unknowns are all raised to the rst power. Let us take, as an example, the general form of a linear equation in three unknowns; thus, ax by cz k 41
where x, y, z denote the values of three unknown quantities, with a, b, c being the corresponding constant coe cients of the unknowns, and k denoting a single constant term on the right-hand side. In our problems the value of the constants will be known, and we will be required to nd the values of the unknown quantities. As a general principle we know that the more complicated a problem is, the greater is the amount of information needed to solve the problem. For the case of linear simultaneous equations, this simply means that the greater the number of unknown values that must be found, the greater is the number of equations that will be required to nd the values. This can be stated as follows. A problem involving linear equations in n unknowns requires, in general, n independent equations for its solution.* Thus, if a problem involves linear equations in two unknowns, then two equations are required to nd a solution; or, if a problem involves linear equations in three unknowns, then three equations are required to nd a solution; and so on. With the above in mind, we now wish to introduce a general procedure that can be used to nd the solution to any system of n simultaneous linear equations. To do this, let us take, as an example, the solving of a set of three simultaneous linear equations, in which we ll denote the values of the three unknowns by x, y, and z. Doing this will clearly show why the procedure is correct, and why it is valid for any set of n such equations. We proceed as follows. Let the three simultaneous equations be designated as eq. (42), in which a through i represent constant coe cients and where P, Q, and R are constant terms on the right-hand sides of the equations, as shown. (To reduce the writing time we ll not use the usual double-subscript notation in this discussion.)
* Two equations are independent if neither one can be derived from the other. As a simple example, x y 10 and 2x 2y 20 are not independent, because the second one was derived from the rst by multiplying it by 2.
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