Linear Algebra

298 Chapter 4. Determinants
1.10 Show that the equation of a line in R 2 thru (x1,y1) and(x2,y2) is expressed
by this determinant.
�
x y
�
1
det(
)=0 x1 �= x2
x1 y1 1
x2 y2 1
� 1.11 Many people know this mnemonic for the determinant of a 3×3 matrix: first
repeat the first two columns and then sum the products on the forward diagonals
and subtract the products on the backward diagonals. That is, first write
�
and then calculate this.
� h1,1 h1,2 h1,3 h1,1 h1,2
h2,1 h2,2 h2,3 h2,1 h2,2
h3,1 h3,2 h3,3 h3,1 h3,2
h1,1h2,2h3,3 + h1,2h2,3h3,1 + h1,3h2,1h3,2
−h3,1h2,2h1,3 − h3,2h2,3h1,1 − h3,3h2,1h1,2
(a) Check that this agrees with the formula given in the preamble to this section.
(b) Does it extend to other-sized determinants?
1.12 The cross product of the vectors
� � � �
x1
y1
�x =
�y =
is the vector computed as this determinant.
�
�e1 �e2
�
�e3
�x × �y =det(
)
x2
x3
y2
y3
x1 x2 x3
y1 y2 y3
Note that the first row is composed of vectors, the vectors from the standard basis
for R 3 . Show that the cross product of two vectors is perpendicular to each vector.
1.13 Prove that each statement holds for 2×2 matrices.
(a) The determinant of a product is the product of the determinants det(ST)=
det(S) · det(T ).
(b) If T is invertible then the determinant of the inverse is the inverse of the
determinant det(T −1 )=(det(T )) −1 .
Matrices T and T ′ are similar if there is a nonsingular matrix P such that T ′ =
PTP −1 . (This definition is in Chapter Five.) Show that similar 2×2 matrices have
the same determinant.
� 1.14 Prove that the area of this region in the plane
� �
x2
is equal to the value of this determinant.
Compare with this.
y2
det(
det(
�
x1 x2
y1 y2
�
x2 x1
y2 y1
�
)
�
)
� �
x1
y1