UNIT-7-OBJECTIVE 1- DETERMINANTS; EXPANSION BY MINORS
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There are several ways to find the value of a 3*3 determinant. One way to find
the value of a determinant is to expand a determinant by minors. In general, the
minor of an element in a 3*3 determinant is the 2*2 determinant that is found
by eliminating the row and column that contain that element. The signs for the
expansion of a 3*3 determinant by minors is

The value of the determinant is the sum of the products of the elements in the
choosen row and column, each with the appropriate  sign and their minors.

Example 1

You can expand by any row or column. We'll expand using elements in the second
row and their minors.

Sign of the element

If we expand by third column we would get the following :

Expansion by minor can be used with any determinant of order three or greater.

Determinants can be used to solve systems of equations. To solve 3 equations and
3 unknowns the denominator for all three variables is the determinant of the
coefficient of these variables which is a 3*3 determinant. For each variables
numerator replace the coefficients for that variable with the constants on the
right of the equations.

Example2

We can substitute the value for x and y into any of the original equation to
solve for z or we can evaluate the determinant.
(Choosing the first equation)