MATH 3912 - Assignment 1
1. Compute the determinant of
6 0 −8
8 3 0
6 3 −4
2. Compute the determinant of the lower triangular matrix
a11 0 0 0
a21 a22 0 0
a31 a32 a33 0
a41 a42 a43 a44
3. Suppose Ais a lower triangular nby nmatrix as defined below. Show that det(A) = Qn
i=1 aii, or in
other words the determinant of Ais the product of the entries in the main diagonal.
A=
a11 0. . . 0
a21 a22 0. . . 0
a31 a32 a33 0. . . 0
.
.
........
.
.
an1an2an3. . . ann
4. Find the determinant of the matrix Vdefined as follows:
V=
1z0z2
0
1z1z2
1
1z2z2
2
5. Solve the following system of equations, if there is a solution:
8x1−2x2+x3= 1
2x1−8x2= 7
−2x1−6x2−7x3= 2
6. Suppose p(x) = a0+a1x+a2x2is a quadratic polynomial. Find the coefficients of that polynomial so
that
p(−1) = 1
p(0) = 4
p(1) = 1
7. Suppose
~v1=h7,−5,5i,~v2=h6,0,0i, and ~v3=h−3,5,0i
are three vector in R3. Are they linearily independent? What about the vectors
~w1=h−3,−6,−7i,~w2=h0,2,−15i, and ~w3=h3,8,−8i
How about
~z1=h6,0,0i,~z2=h−3,5,0i,~z3=h−3,−6,−7i, and ~z4=h0,2,−15i
