Linear Algebra

332 Chapter Four. DeterminantsThe second is defined by the vectors x 1( 13)and( 21), and one of the properties ofthe size function — the determinant — is that therefore the size of the secondbox is x 1 times the size of the first box. Since the third box is defined by thevector x 1( 13)+ x2( 21)=( 68)and the vector( 21), and since the determinant doesnot change when we add x 2 times the second column to the first column, thesize of the third box equals that of the second.∣ 6 2∣∣∣∣ ∣8 1∣ = x 1 · 1 2x 1 · 3 1∣ = x 1 ·1 2∣3 1∣Solving gives the value of one of the variables.6 2∣8 1∣x 1 =1 2∣3 1∣= −10−5 = 2The generalization of this example is Cramer’s Rule: if |A| ≠ 0 then thesystem A⃗x = ⃗b has the unique solution x i = |B i |/|A| where the matrix B i isformed from A by replacing column i with the vector ⃗b. The proof is Exercise 3.For instance, to solve this system for x 2⎛ ⎞ ⎛1 0 4⎜ ⎟ ⎜x ⎞ ⎛ ⎞1 2⎟ ⎜ ⎟⎝2 1 −1⎠⎝x 2 ⎠ = ⎝ 1⎠1 0 1 x 3 −1we do this computation.1 2 42 1 −1∣1 −1 1∣x 2 =1 0 42 1 −1∣1 0 1∣= −18−3Cramer’s Rule allows us to solve simple two equations/two unknowns systemsby eye (they must be simple in that we can mentally compute with the numbersin the system). With practice a person can also do simple three equations/threeunknowns systems. But computing large determinants takes a long time sosolving large systems by Cramer’s Rule is not practical.Exercises1 Use Cramer’s Rule to solve each for each of the variables.