(iii) If two rows or two columns are identical the determinant is zero (follows from (ii)).
(iv) Any factors common to a row (column) may be factored out from the determinant.
(v) A row may be increased or decreased by equal multiples of another row (similarly,
for columns) without changing the determinant.
(vi) A determinant in which a row or a column consists of the sum or difference of two or
more terms, may be expanded as the sum or difference of two or more determinants.(vii) The determinant of the product of two (square) matrices is the product of the deter-
minants of the factors.
Problem 13.10
Illustrate the above properties of determinants by applying them to general
2 ×2 determinants.(i)∣
∣
∣
∣ab
cd∣
∣
∣
∣=ad−bc=∣
∣
∣
∣ac
bd∣
∣
∣
∣(ii)∣
∣
∣
∣ab
cd∣
∣
∣
∣=ad−bc=−(bc−ad)=−∣
∣
∣
∣ba
dc∣
∣
∣
∣
(iii) If two rows (columns) are the same then exchanging them will not change the
determinant at all, but by (ii) it will change the sign. So the determinant will be its
own negative and must therefore be zero. Or, directly:
∣
∣
∣
∣aa
bb∣
∣
∣
∣=ab−ab=^0(iv)∣
∣∣
∣ka b
kc d∣
∣∣
∣=kad−kbc=k(ad−bc)=k∣
∣∣
∣ab
cd∣
∣∣
∣(v)∣
∣
∣
∣a+kb b
c+kd d∣
∣
∣
∣=(a+kb)d−b(c+kd)=ad−bc=∣
∣
∣
∣ac
bd∣
∣
∣
∣(vi)∣
∣
∣
∣a+eb
c+f d∣
∣
∣
∣=(a+e)d−b(c+f)=ad−bc+ed−bf=∣
∣
∣
∣ab
cd∣
∣
∣
∣+∣
∣
∣
∣eb
fd∣
∣
∣
∣(vii)
[
ab
cd][
ef
gh]
=[
ae+bg af+bh
ce+dg cf+dh]Now ∣
∣
∣
∣
ae+bg af+bh
ce+dg cf+dh∣∣
∣
∣=(ae+bg)(cf+dh)−(af+bh)(ce+dg)=aecf+aedh+bgcf+bgdh−af ce−af dg
−bhce−bhdg