S OCIAL E COLOGY W EEK 8: S AMPLE M IDTERM 2 Q UESTIONS
M ATH 121A W INTER 2017
1. Find the value of k that satisfies the following equation.
0
1
0
1
b1 + 3c1 b2 + 3c2 b3 + 3c3
a1 a2 a3
det @ 2a1 + c1 2a2 + c2 2a3 + c3 A = k det @ b1 b2 b3 A
2c1
2c2
2c3
c1 c2 c3
2. (This problem should require almost no computation.)
a. Find a matrix A satisfying the following
0
1 0
1 0
1
1 0 0
1 2 3
1 2 3
@ 1 1 0 AA@ 4 5 6 A = @ 3 3 3 A
0 0 1
7 8 9
3 3 3
b. For each of the four matrices in the above equation, determine its rank.
3. Give an example or prove none exists.
a. Two elementary matrices such that their product is not an elementary matrix.
b. A linear transformation f : V ! W between finite-dimensional vector spaces which is not
onto and for which rank(f ) = dim(W ).
c. Two 4 ⇥ 4 matrices A, B where det(A) = det(B) but rank(A) 6= rank(B).
4. Using the rank-nullity theorem, prove that the system of equations
a1 x 1 + a2 x 2 + a3 x 3 = 0
b1 x 1 + b2 x 2 + b3 x 3 = 0
always has infinitely many solutions (x1 , x2 , x3 ) 2 R3 .
5. Answer True or False to each of the following, and briefly explain your answers.
a. If A, B are square matrices such that AB = In , then A is invertible.
b. Every elementary matrix is a square matirx.
✓
◆
✓
◆
a b
ka kb
c. If
is an invertible matrix, and if k 2 R is non-zero, then
is also an
c d
c d
invertible matrix.