Linear Algebra, 2020a

Section I. Definition 335
could possibly return, we haven’t yet shown that such a function exists for all n.
The rest of this section does that.
Exercises
For these, assume that an n×n determinant function exists for all n.
̌ 2.8 Find each determinant by performing one row operation.
1 −2 1 2
(a)
2 −4 1 0
1 1 −2
0 0 −1 0
(b)
0 0 4
∣
∣
0 0 0 5
∣ 0 3 −6∣
̌ 2.9 Use Gauss’s Method to find each determinant.
1 0 0 1
3 1 2
(a)
3 1 0
(b)
2 1 1 0
∣0 1 4∣
−1 0 1 0
∣
1 1 1 0
∣
2.10 Use Gauss’s Method to find each.
(a)
∣ 2 −1
1 1 0
−1 −1∣
(b)
3 0 2
∣5 2 2∣
2.11 For which values of k does this system have a unique solution?
x + z − w = 2
y − 2z = 3
x + kz = 4
z − w = 2
̌ 2.12 Express each of these in terms of |H|.
∣ h 3,1 h 3,2 h 3,3∣∣∣∣∣
(a)
h 2,1 h 2,2 h 2,3
∣h 1,1 h 1,2 h 1,3 ∣ −h 1,1 −h 1,2 −h 1,3 ∣∣∣∣∣
(b)
−2h 2,1 −2h 2,2 −2h 2,3
∣−3h 3,1 −3h 3,2 −3h 3,3 ∣ h 1,1 + h 3,1 h 1,2 + h 3,2 h 1,3 + h 3,3∣∣∣∣∣
(c)
h 2,1 h 2,2 h 2,3
∣ 5h 3,1 5h 3,2 5h 3,3
̌ 2.13 Find the determinant of a diagonal matrix.
2.14 Describe the solution set of a homogeneous linear system if the determinant of
the matrix of coefficients is nonzero.
̌ 2.15 Show that this determinant is zero.
y + z x+ z x+ y
x y z
∣ 1 1 1 ∣
2.16 (a) Find the 1×1, 2×2, and 3×3 matrices with i, j entry given by (−1) i+j .
(b) Find the determinant of the square matrix with i, j entry (−1) i+j .
2.17 (a) Find the 1×1, 2×2, and 3×3 matrices with i, j entry given by i + j.