A 23 =(− 1 )^2 +^3∣
∣
∣
∣a 11 a 12
a 31 a 32∣
∣
∣
∣Then we can easily check that
3 =a 21 A 21 +a 22 A 22 +a 23 A 23Similarly, we can check that
3 =a 13 A 13 +a 23 A 23 +a 33 A 33(expansion by the third column).
You may have spotted that the example equations in Problem 13.8 can be solved much
more simply than by employing determinants. Adding them gives
4 x 1 = 3yieldingx 1 immediately and thenx 2 can be obtained from one of the equations. So
this already appears to be a sledgehammer to crack a nut – why bother with determi-
nants at all? Because they provide a systematic, ‘easily’ remembered, and automatic
means of writing down the solution of any system of linear equations no matter how
complicated. The determinantal form does not depend on accidental properties of the
coefficients.
Problem 13.9
Evaluate∣
∣
∣
∣
∣31 − 2
102
− 3 − 14∣
∣
∣
∣
∣Expand by the second row because it contains a zero:
= 1 (− 1 )∣
∣
∣
∣1 − 2
− 14∣
∣
∣
∣+^0∣
∣
∣
∣3 − 2
− 34∣
∣
∣
∣+^2 (−^1 )∣
∣
∣
∣31
− 3 − 1∣
∣
∣
∣=−( 4 − 2 )+ 0 =− 2Note that the final determinant is zero because the two rows (or columns) are proportional
to each other – see below.
Above we have both defined determinants and given a method for evaluating them. We
now need their properties. These are best illustrated using 2×2examples.
(i) A determinant and its transpose (i.e. the determinant obtained from it by
transposing rows into columns – cf: transpose of a matrix (381➤
)) have the same
value.(ii) Interchanging two rows or two columns changes the sign of the determinant.