To calculate a determinant, we rearrange terms using the distributive property.
(1)Each quantity in parentheses represents a determinant that is the part of
the determinant remaining when the row and column of the multiplier are
eliminated, as shown below.
Eliminate the 1strow
and 1stcolumn.Eliminate the 2ndrow
and 1stcolumn.Eliminate the 3rdrow
and 1stcolumn.These determinants are called minorsof the elements in the determi-
nant. In the determinant above, the minors of and are, respectively,
MinorsOBJECTIVE 2 Use expansion by minors to evaluate 3 3 determinants.
We evaluate a determinant by multiplying each element in the first column by
its minor and combining the products as indicated in equation (1). This procedure is
called expansion of the determinant by minorsabout the first column.
Evaluating a 3 3 DeterminantEvaluate the determinant using expansion by minors about the first column.
(^3)
1
- 1
1
3
- 2
1
- 2
- 3
2
3
EXAMPLE 2 :
3 * 3
:
`
b 2
b 3
c 2
c 3
,
b 1
b 3
c 1
c 3
,and
b 1
b 2
c 1
c 2
`.
a 1 ,a 2 , a 3
2 2 3 3
a 31 b 1 c 2 - b 2 c 12 3
a 1
a 2
a 3
b 1
b 2
b 3
c 1
c 2
c 3
3
a 21 b 1 c 3 - b 3 c 12 3
a 1
a 2
a 3
b 1
b 2
b 3
c 1
c 2
c 3
3
a 11 b 2 c 3 - b 3 c 22 3
a 1
a 2
a 3
b 1
b 2
b 3
c 1
c 2
c 3
3
3 * 3
2 * 2
(^3)
a 1
a 2
a 3
b 1
b 2
b 3
c 1
c 2
c 3
(^3) =a
11 b 2 c 3 - b 3 c 22 - a 21 b 1 c 3 - b 3 c 12 +a 31 b 1 c 2 - b 2 c 12
3 * 3
716 APPENDIX A Determinants and Cramer’s Rule
Value of a 3 3 Determinant(^3)
a 1
a 2
a 3
b 1
b 2
b 3
c 1
c 2
c 3
(^3) 1 a