Determinants and
Cramer’s Rule
APPENDIX
A
715
Recall from Section 4.4that an ordered array of numbers within square brackets is
called a matrix(plural matrices). Matrices are named according to the number of
rows and columns they contain. A square matrixhas the same number of rows and
columns.
ColumnsRowsAssociated with every square matrixis a real number called the determinantof
the matrix. A determinant is symbolized by the entries of the matrix placed between
two vertical lines.
Like matrices, determinants are named according to the number of rows and columns
they contain.
OBJECTIVE 1 Evaluate 2 2 determinants.As mentioned above, the value
of a determinant is a real number.We use the following rule to evaluate a
determinant.
2 * 2
:
(^3)
7
0
6
4
1
0
3
5
1
` 3
2
7
3
1
`
c
- 1
1
0
- 2
c d
2
7
3
1
5
2
d
OBJECTIVES
1 Evaluate 2 2
determinants.
2 Use expansion by
minors to evaluate
3 3 determinants.
3 Understand the
derivation of
Cramer’s rule.
4 Apply Cramer’s rule
to solve linear
systems.:
:
Matrix2 * 3Value of a 2 2 Determinant`
a
c
b
d
` adbc
:
square matrix2 * 2determinant2 * 2
determinant3 * 3Evaluating a 2 2 DeterminantEvaluate the determinant.
Here and so
NOW TRYA 3 * 3 determinant is evaluated in a similar way.
`
- 1
4
- 3
- 2
` =- 11 - 22 - 1 - 324 = 2 + 12 = 14.
a=-1,b=-3,c=4, d=-2,
`
- 1
4
- 3
- 2
`
EXAMPLE 1 :
NOW TRY
EXERCISE 1
Evaluate the determinant.
NOW TRY ANSWER
- 42
`^
63
4 - 5
(^) `