Linear Algebra

302 Chapter 4. Determinants
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2.8 Use Gauss’ method to find each.
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(b) �3
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2.9 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
� 2.10 Expresseachoftheseintermsof|H|.
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h3,2 h3,3�
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(a) �h2,1
h2,2 h2,3�
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h1,1 h1,2 h1,3
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(b) �−2h2,1
−2h2,2 −2h2,3�
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−3h3,1 −3h3,2 −3h3,3
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+ h3,1 h1,2 + h3,2 h1,3 + h3,3�
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(c) � h2,1 h2,2 h2,3 �
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5h3,1 5h3,2 5h3,3
� 2.11 Find the determinant of a diagonal matrix.
2.12 Describe the solution set of a homogeneous linear system if the determinant
of the matrix of coefficients is nonzero.
� 2.13 Show that this determinant is zero.
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2.14 (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 .
2.15 (a) Find the 1×1, 2×2, and 3×3 matrices with i, j entry given by i + j.
(b) Find the determinant of the square matrix with i, j entry i + j.
� 2.16 Show that determinant functions are not linear by giving a case where |A +
B| �= |A| + |B|.
2.17 The second condition in the definition, that row swaps change the sign of a
determinant, is somewhat annoying. It means we have to keep track of the number
of swaps, to compute how the sign alternates. Can we get rid of it? Can we replace
it with the condition that row swaps leave the determinant unchanged? (If so then
we would need new 1 ×1, 2×2, and 3×3 formulas, but that would be a minor
matter.)
2.18 Prove that the determinant of any triangular matrix, upper or lower, is the
product down its diagonal.
2.19 Refer to the definition of elementary matrices in the Mechanics of Matrix
Multiplication subsection.
(a) What is the determinant of each kind of elementary matrix?