Exercises 8.6 Exercises
Minors and cofactors.
In each case, calculate the requested minors and cofactors of the given matrix. For the larger matrices in Exercise 5 and Exercise 6, express the requested minors as unevaluated determinant matrices and cofactors in terms of the minor.
2.
(a)
Answer.
(b)
Answer.
(c)
Answer.
(d)
Answer.
3.
(a)
Answer.
(b)
Answer.
(c)
Answer.
(d)
Answer.
(e)
Answer.
(f)
Answer.
(g)
Answer.
(h)
Answer.
4.
(a)
Answer.
(b)
Answer.
(c)
Answer.
(d)
Answer.
(e)
Answer.
(f)
Answer.
(g)
Answer.
(h)
Answer.
5.
(a)
Answer.
(b)
Answer.
(c)
Answer.
(d)
Answer.
6.
(a)
Answer.
(b)
Answer.
(c)
Answer.
(d)
Answer.
Cofactor expansions.
In each case, express the requested cofactor expansion as a linear combination of minor determinants, bringing the cofactor signs to the front of each term. For matrices larger than do not evaluate the minor determinants.
7.
(a)
Along the first row.
Answer.
(b)
Along the second row.
Answer.
8.
(a)
Along the first column.
Answer.
(b)
Along the second column.
Answer.
9.
(a)
Along the third row.
Answer.
(b)
Along the second column.
Answer.
(c)
Along the third column.
Answer.
10.
(a)
Along the first row.
Answer.
(b)
Along the first column.
Answer.
(c)
Along the second column.
Answer.
(d)
Along the second row.
Answer.
11.
(a)
Along the first column.
Answer.
(b)
Along the second row.
Answer.
(c)
Along the third column.
Answer.
(d)
Along the fourth row.
Answer.
12.
(a)
Along the first row.
Answer.
(b)
Along the second column.
Answer.
(c)
Along the third row.
Answer.
(d)
Along the fourth column.
Answer.
Choosing a cofactor expansion.
For each matrix, choose a row or column for which calculating the cofactor expansion would require the fewest minor determinant calculations.
13.
Answer.
A cofactor expansion along either the first column or the second row would involve only two minor determinant calculations, rather than three.
14.
Answer.
A cofactor expansion along the third row would involve only one minor determinant calculation, rather than three.
15.
Answer.
A cofactor expansion along any of the first or second rows or the first or fourth columns would involve only three minor determinant calculations, rather than four.
16.
Answer.
A cofactor expansion along the second column would involve only two minor determinant calculations, rather than four.
Computing determinants.
Compute the determinant of each matrix.
17.
Answer.
18.
Answer.
19.
Solution.
20.
Solution.
21.
Solution.
22.
Solution.
23.
Solution.
Let represent the matrix. Expanding along the first row, we have
To compute the minor determinants in this cofactor expansion we have used the general formula for the determinant of a matrix. (See Subsection 8.3.3.)
24.
Solution.
Let represent the matrix. Expand along the second row so that there are only two cofactors to compute:
To compute the minor determinants in this cofactor expansion we have used the general formula for the determinant of a matrix. (See Subsection 8.3.3.)
25.
Solution.
Let represent the matrix. Choose the second column for a cofactor expansion since it will involve the fewest minor determinant calculations:
Let’s compute these two minor determinants separately; call the first matrix and the second Neither has any zero entries, but the third row in each contains a couple of ones, so we’ll choose to expand along that row in each. In both expansions, we will use the general formula for the resulting minor determinants (see Subsection 8.3.3).
Substituting these results into our original cofactor expansion for we have
26.
Solution.
Let represent the matrix. Any row or column we choose will contain two zeros, so just expand along the first row:
Let’s compute these two minor determinants separately; call the first matrix and the second The second row in each contains a couple of zeros, so we’ll choose to expand along that row in each. In both expansions, we will use the general formula for the resulting minor determinants (see Subsection 8.3.3).
Substituting these results into our original cofactor expansion for we have
27.
Solution.
Let represent the matrix. Notice that is lower triangular, so by Statement 1 of Proposition 8.5.2 its determinant is simply the product of its diagonal entries:
28.
Solution.
Let represent the matrix. Notice that is upper triangular, so by Statement 1 of Proposition 8.5.2 its determinant is simply the product of its diagonal entries:
29.
Solution.
Let represent the matrix. Notice that is diagonal, so by Statement 1 of Proposition 8.5.2 its determinant is simply the product of its diagonal entries:
30.
Solution.
Let represent the matrix. Notice that is scalar, so by Statement 2 of Proposition 8.5.2 its determinant is simply the power of the common diagonal entry value, with exponent equal to the size of the square matrix:
31. Counterintuitive determinant examples.
Provide example matrices with the requested properties.
- A
matrix the entries of which are all nonzero, but for which - A nonzero
upper triangular matrix for which - A nonzero
diagonal matrix for which - A pair of
matrices and for which both and but (This demonstrates that is not equal to in general.)
Variable determinants.
Each matrix below contains one or more entries involving a parameter. Compute the determinant as a formula in that parameter. Then determine all values of the parameter for which the determinant is equal to
32.
Solution.
This determinant will equal when
33.
Solution.
This determinant will equal when
34.
Solution.
Let represent the matrix. Expanding along the second row so that there are is only cofactor to compute:
This determinant will equal when
35.
Solution.
Let represent the matrix. This is an upper triangular matrix, so by Statement 1 of Proposition 8.5.2 its determinant is simply the product of its diagonal entries:
This determinant will equal zero when is equal to one of or