Math 369 HW #2
Due at the beginning of class on Friday, September 12
Reading: Sections MM, HSE, MISLE, LC.
Problems:
1. Consider the matrices
A=
1 2 1
−2 3 2


−1 2
and B =  3 −4
−3 −2
(a) Compute AB.
(b) Compute BA.
2. Given a square matrix A = (aij ) ∈ Matn,n (R), the trace of A, denoted tr(A), is the sum of
the diagonal elements of A. For example, if A is 2 × 2, then
a11 a12
tr(A) = tr
= a11 + a22 .
a21 a22
Now, suppose A = (aij ) and B = (bij ) are both 2 × 2 matrices.
(a) Is it true that tr(A + B) = tr(A) + tr(B)? If yes, prove it using only the definitions of
trace and matrix addition. If no, give a counterexample.
(b) Is it true that tr(AB) = tr(A) tr(B)? If yes, prove it using only the definitions of trace
and matrix multiplication. If no, give a counterexample.
3. For any square matrix A ∈ Matn,n (R) and any natural number N , define the matrix power
AN = A
| · A{z· · · A}.
n times
Let D =
2 0
0 5
1 4
and let U =
.
0 1
(a) Compute D2 and D3 .
(b) What is DN ?
(c) Compute U 2 and U 3 .
(d) What is U N ?
*
*
4. Suppose that u and v are solutions to the homogeneous equation
*
*
A x = 0.
*
*
Prove that u + v is also a solution to this equation.
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5. Our three row operations on matrices can be realized as multiplication (on the left) by certain
special elementary matrices. For example, to swap the first and third rows ina 3 × 3 matrix,
we can multiply the matrix on the left by


0 0 1
0 1 0  .
1 0 0
As you can easily check,


 

0 0 1
1 2 3
7 8 9
0 1 0 4 5 6 = 4 5 6 .
1 0 0
7 8 9
1 2 3
(a) What matrix corresponds to multiplying row 2 of a 3 × 3 matrix by λ?
(b) What matrix corresponds to replacing row 1 of a 3 × 3 matrix with row 1 plus β times
row 3?
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