Linear Algebra, 2020a

330 Chapter Four. Determinants
⎛
⎞ ⎛
⎞
1 −1 3 1 −1 3
(c) A = ⎝2 2 −6⎠, B = ⎝1 1 −3⎠
1 0 4 1 0 4
̌ 1.7 Find the determinant of this 4×4 matrix by following the plan: perform Gauss’s
Method and look for the determinant to remain unchanged on a row combination,
to change sign on a row swap, to rescale on the rescaling of a row, and such that
the determinant of the echelon form matrix is the product down its diagonal.
⎛
⎞
1 2 0 2
⎜2 4 1 0
⎟
⎝0 0 −1 3⎠
3 −1 1 4
1.8 Show this. ⎛
⎞
1 1 1
det( ⎝ a b c⎠) =(b − a)(c − a)(c − b)
a 2 b 2 c 2
̌ 1.9 Which real numbers x make this matrix singular?
( )
12 − x 4
8 8− x
1.10 Do the Gaussian reduction to check the formula for 3×3 matrices stated in the
preamble to this section.
⎛ ⎞
a b c
⎝d e f⎠ is nonsingular iff aei + bfg + cdh − hfa − idb − gec ≠ 0
g h i
1.11 Show that the equation of a line in R 2 through (x 1 ,y 1 ) and (x 2 ,y 2 ) is given by
this determinant. ⎛
x y
⎞
1
det( ⎝x 1 y 1 1⎠) =0 x 1 ≠ x 2
x 2 y 2 1
1.12 Many people have learned this mnemonic for the determinant of a 3×3 matrix:
copy the first two columns to the right side of the matrix, then take the
products down the forward diagonals and add them together, and then take the
products on the backward diagonals and subtract them. That is, first write
⎛
⎞
h 1,1 h 1,2 h 1,3 h 1,1 h 1,2
⎝h 2,1 h 2,2 h 2,3 h 2,1 h 2,2
⎠
h 3,1 h 3,2 h 3,3 h 3,1 h 3,2
and then calculate this.
h 1,1 h 2,2 h 3,3 + h 1,2 h 2,3 h 3,1 + h 1,3 h 2,1 h 3,2
−h 3,1 h 2,2 h 1,3 − h 3,2 h 2,3 h 1,1 − h 3,3 h 2,1 h 1,2
(a) Check that this agrees with the formula given in the preamble to this section.
(b) Does it extend to other-sized determinants?
1.13 The cross product of the vectors
⎛ ⎞ ⎛ ⎞
x 1
y 1
⃗x = ⎝x 2
⎠ ⃗y = ⎝y 2
⎠
x 3 y 3