Winter 2020 Math 21 Practice Final Question 1. Let
1 2 −1 A=2 2 4.
Express the square matrix A and its inverse as a product of elementary matrices.
Question 2. Let
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a. Calculate C23(A). b. Calculate det(A).
2 3 4 A=−1 0 2.
Question 3. Let A, B, C, D be 3 × 3 matrices such that det(A) = 5, det(B) = −3, det(C) = 10, det(D) = 4.
a. Calculate det(ABC). b. Calculate det(AB2C).
c. Calculate det(ABC−1) d. Calculate det(ABCDTr).
e. Calculate det(2A2B−1CTrD)
f. Calculate the determinat of 0 1 0 D.
Question 4. Let A = 1 3 −1. Find the inverse of A using
Winter 2020 Math 21 Practice Final
Question 5. Solve the following system using the Cramer’s rule.
x+y+z=2 6x − 4y + 5z = 31. 5x + 2y + 2z = 13
11 13 32 Question6.Letm1= −1 −1 ,m2= −2 −2 ,m3= −2 −3 ,m=
0 h . For which values of h we have m ∈ Span(m1, m2, m3)?
Question 7. Let
p1(x)=x2 +2x+1,p2(x)=2×2 +3x+1,p3(x)=2×2,p4(x)=2×2 +x+1.
Find a basis of the Span(p1(x), p2(x), p3(x), p4(x)). Question 8. Consider the basis
B={1+x,1+x2,x−x2 +2×3,1−x−x2}
of R3[x]. Find the coordinate vector of p(x) = 1+2x+3×2 +4×3 with respect to the basis B.
Question 9. Let A be a 6 × 4 matrix and assume that A has 2 pivot columns. Find nullity(A) and rank(A).
1 0 −1 2 Question10.LetA=2 1 2 3.
−1 0 1 2 a. Find a basis for the column space of A.
b. Find the rank of A.
c. Find a basis for the null space of A.
d. Find the nullity of A.
Question 11. Let T : M2×2(R) → R3[x] be the linear transformation
T c d = 2a+(b−d)x−(a+c)x +(a+b−c−d)x .
ab 2 3 a. Find a basis for the range of T.
Winter 2020 Math 21 Practice Final
b. Find the rank of T. c. Is T surjective?
d. Find a basis for the kernel of T . e. Find the nullity of T.
f. Is T injective?
Question 12. Let B = (p1(x), p2(x), p3(x)) where
p1(x) = 1 − x, p2(x) = 1 − x2, p3(x) = 1 − 2x + 2×2. 1
B is a basis of R2(x). Let [p(x)]B = 2. Find p(x). 3
Question 13. Let V be a 3-dimensional vector space. Consider the bases B={v1,v2,v3}andB′ ={w1,w2,w3}suchthat
v1 =w1 +3w2 +3w3 v2 = w1 + 4w2 + 3w3. v3 =w1 +3w2 +4w3
Find the basis transition matrices PB→B′ and PB′→B.
Question 14. Let V be a 3-dimensional vector space with bases B =
(v1,v2,v3) and B′ = (w1,w2,w3). Let PB→B′ = 3 3 1 and v = v1 +v2 +
v3. Find [v]B′ .
Question 15. Let V be a 3-dimensional vector space with bases B =
5 3 4 (v1,v2,v3),B′ =(w1,w2,w3),andB∗ =(u1,u2,u3). LetPB→B′ = 1 −3 2
and PB′→B∗ = 3 −3 −1. Find PB→B∗ . 1 −2 3
Question 16. Let A be a 5 × 5 matrix with 2 pivot columns. a. Find det(A).
b. Find rank(A).
Winter 2020 Math 21 Practice Final c. Find nullity(A).
Question 17. Let T : R4 → R3 be a linear transformation and suppose that the standard matrix of T has 2 pivot columns.
a. Find rank(T). b. Is T surjective?
c. Find nullity(T). d. Is T injective?
Question 18. Let T : R3[x] → M2×2(R) be a linear transformation such
2 3 a+c −a+d T(a+bx+cx +dx )= b−c −b−d .
Find the matrix of T with respect to the standard bases of R3[x] and M2×2(R). 2 −3 0
Question 19. Let 2 −5 0. Find all eigenvalues, and eigenspaces 003
of A. Is the matrix A diagonalizable?
Question 20. Let A = −5 3 a . Find all values of a the matrix 4 −2 −1
A has eigenvalues 0, 3, and −3.
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