Linear Algebra Quiz 2
Problem 1. (2 points each) For a square matrix A, answer with True or False.
(1) The matrix A is invertible if and only if the sum of eigenvalues is not zero.
(2) If nullity(A) > 0, then A is diagonalisable.
(3) If A is orthogonal matrix, then A−1 is orthogonal matrix.
(4) If A is orthogonally diagonalisable, then AT = A.
Sol. (1) F (2) F (3) T (4) T

0
Problem 2. (2+3 points) Let A =  1
0
0
0
1

−2
1 .
2
(1) Find all eigenvalues of A.
(2) Find an invertible matrix P such that P −1 AP is a diagonal matrix.

 
 

2
−2
−1
Sol. (1) -1,1,2 (2) Columns are  −3 ,  −1 ,  0 
1
1
1
Problem 3. (4 points) Let V be a 4-dimensional vector space and B = {v1 , v2 , v3 , v4 } be a basis for V . Find an orthonormal
basis for V .
Sol. w1 = ||v11 || v1 , ||v11 || v1 , w2 = v2 − ⟨v2 , w1 ⟩, w3 = v3 − ⟨v3 , w1 ⟩ − ⟨v3 , w2 ⟩, w4 = v4 − ⟨v4 , w1 ⟩ − ⟨v4 , w2 ⟩ − ⟨v4 , w3 ⟩.

3
Problem 4. (5 points) Let A =  4
−3

0 −1
2 −5 .
1 4
(1) For x = (x1 , x2 , x3 )T , express xT Ax in a quadratic form q(x).
(2) Determine the definiteness of q(x).


3
2 −2
2 −2  x (2) All principal minors are positive so it is positive definite.
Sol. (1) q(x) = xT  2
−2 −2 4
Problem 5. (3 points) Let {ai } and {bi } be sequences of real numbers such that an+1 = an + bn and bn+1 = an for n ≥ 0,
a0 = 1, and b0 = 0. Find the explicit form of ai and bi .
1
Sol. ai = √15
√ i+1
√ i+1
1+ 5
1− 5
−
2
2
√ √ i i
1+ 5
1− 5
, bi = √15
−
2
2
2