det(
A
−
λ
I
2
) = 0
A
=
8
−
4
7
−
1
λ
=
(No Response)
det(
A
−
λ
I
2
) = det
−
λ
= det
=
λ
2
−
7
λ
+
20
8
−
4
7
−
1
1 0
0 1
8
−
λ
−
7
−
λ
−
1
(
−
7
)
2
−
4(1)(
20
) =
−
31
< 0.
= 0
4
10.
–/3 pointsHoltLinAlg1 5.1.050.
Determine all real values of
λ
such the
for the given matrix
A
. (Enter your answers as
a comma-separated list. If an answer does not exist, enter DNE.)
λ
λ
=
4
.
7
λ
11.
–/5 pointsHoltLinAlg1 5.1.053.
For each given matrix
A
, first compute det(
A
). Then interchange two rows of your choosing and compute
the determinant of the resulting matrix
A
'.
(a)
(b)
Form a conjecture about the effect of row interchanges on determinants.
Interchanging rows halves the determinant.
Interchanging rows doubles the determinant.
Interchanging rows changes the sign of the determinant.
Interchanging rows does not change the determinant.
Interchanging rows has no consistent effect on the determinant.
Solution or Explanation
A
=
3 4
−
5 2
det(
A
) =
(No Response)
26
det(
A'
) =
(No Response)
-26
A
=
1
3
−
1
2
0
3
0 1
−
1
det(
A
) =
(No Response)
1
det(
A'
) =
(No Response)
-1
det
=
26
; interchanging rows, det
=
−
26
.
3 4
−
5 2
−
5 2
3 4
12.
–/5 pointsHoltLinAlg1 5.1.057.
For each given matrix
A
, first compute det(
A
). Then multiply a row of your choosing by 3 and compute
the determinant of the resulting matrix
A
'.
(a)
(b)
Form a conjecture about the effect on determinants of multiplying a row times a scalar.
Multiplying a row by
c
changes the sign of the determinant.
Multiplying a row by
c
divides the determinant by
c
.
Multiplying a row by
c
has no consistent effect on the determinant.
Multiplying a row by
c
multiplies the determinant by
c
.
Multiplying a row by
c
does not change the determinant.
Solution or Explanation
.
