Unit 5 - Objective 7 - Solving 3 Equations in 3 Unknowns Using Determinants

A 3 x 3 determinant is represented by

= a1 b2 c3 + a3 b1 c2 + a2 b3 c1 - a3 b2 c1 - a1 b3 c2 - a2 b1 c3

where a1, a2, a3, b1, b2, b3, c1, c2, and c3 are any real numbers.  Each element also has a sign associated with it.  Starting with the upper left corner the sign is + 1 and they alternate.

The cofactor of an element in a determinant is the appropriate sign (see above) attached to the determinant obtained by crossing out the row and column of the particular element. This smaller determinant is called the minor of the given element. In the minor of 2 is .

We can use Cramer's rule to solve three equations in 3 unknowns if we have

a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3

then

Providing that:
a1 b2 c3 + a3 b1 c2 + a2 b3 c1 - a3 b2 c1 - a1 b3 c2 - a2 b1 c3 0

Note: The denominators are the same for x, y, and z.

To evaluate a third-order determinant we expand by minors.  For example if we expanded by the first row, multiply each element in the 1st row by its cofactor (sign & minor) then add those 3 values.

= a1 [b2 c3 - b3 c2] - b1 [a2 c3 - a3 c2] + c1 [a2 b3 - a3 b2]

You can expand by any row or column - the sum of the elements in that row (or column) each times its sign and cofactor. These smaller 2 x 2 determinants are evaluated as you did in the last objective.

Examples: