1. Use Cramer's Rule to solve the following matrix equation, show your work.
-1 2 0 a 1
2 1 1 b = 2
-3-1 4 c -1
Solution
o find rules (or formulas) that may be used solve any system of equations, we need to solve the
general system of the form a 1 x + b 1 y = c 1 (1) a 2 x + b 2 y = c 2 (2) We multiply equation (1)
by b 2 and equation (2) by -b 1 and add the right and left hand terms. b 2 a 1 x + b 2 b 1 y = b 2 c
1 (1) - b 1 a 2 x - b 1 b 2 y = - b 1 c 2 (2) ____________________________ b 2 a 1 x - b 1 a 2 x
= b 2 c 1 - b 1 c 2 Assuming that a 1 b 2 - a 2 b 1 is not equal to zero, solve the above equation
for x to obtain. x = ( c 1 b 2 - c 2 b 1 ) / ( a 1 b 2 - a 2 b 1 ) We can use similar steps to eliminate
x and solve for y to obtain. y = ( a 1 c 2 - a 2 c 1 ) / ( a 1 b 2 - a 2 b 1 ) Using the determinant of a
Matrix notation, the solution to the given 2 by 2 system of linear equations is given by x = D x /
D and y = D y / D where D, D x and D y are the determinants defined by For a 3 by 3 system of
linear equations of the form a 1 x + b 1 y + c 1 z = d 1 (1) a 2 x + b 2 y + c 2 z = d 2 (2) a 3 x + b
3 y + c 3 z = d 3 (3) Cramer's rule gives the solution as follows x = D x / D , y = D y / D and z =
D z / D where D, D x, D y and D z are determinants defined by Example: Use Cramer's rule for
a 3 by 3 system of linear equations to solve the following system 2 x - y + 3 z = - 3 - x - y + 3 z =
- 6 x - 2y - z = - 2 Solution: Determinants D, D x, D y and D z are given by (see how to Calculate
Determinant of a Matrix.) D = 21 D x = 21 D y = 42 D z = -21 We now use cramer's rule to find
the solution of the system of equations x = D x / D= 1 y = D y / D= 2 z = D z / D= -1 As an
exercise check that (1,2,-1) is a solution to the given system of equations.