Linear Algebra Answers

Determine for which value (s) of k will the matrix below be non-singular.

(9.1) A = 2−k −3

2 k+1

(9.2) A = 2 2 1

3 1 3

1 3 k

Find A and write the following linear system in the matrix equation form (λI2 − A)X = 0

−x + y = λx

3x + y = λy

For the systems above, find:

(10.1) The determinant (known as the characteristic equation).

(10.2) Solve for λ when det(λI2 − A) = 0.

(10.3) Substitute for each value of λ from (ii) into the equation (λI2 − A)X = 0 and solve the corresponding system for X = x

y

Write the vector u = (1, −2, 5) ∈ R 3 as a linear combination of the vectors u1= (1, 1, 1), u2=

(1, 2, 3), u3= (2, −1, 1)

Show that the vector v = (2, −5, 3) ∈ R 3

cannot be expressed as a linear combination of the

vectors v1= (1, −3, 2), v2= (2, −4, −1), v3= (1, −5, 7).

Suppose that A,B,C and D are matrices with the following sizes: A(5×2),B(4×2),C(4×5),D(4×5)

Determine in each of the following case whether a product is defined.if is it so give the size of the resulting matrix.

i. DC.

ii. -CA+B.

iii. CD-D

5.3 Assume that A and B are matrices of the same size.Determine an expression for A if 2A-B=5(A+2B).

Let A= 1 0 -2 - 1 2 -1 1 3 , and B = 8 3 0 1 4 -7 -5 2 6 Compute A--,(BT)- and B-1A-1. What do you observe about (7.1) (A-1)- in relation to A. (2) (7.2) ((BT)-1)" in relation to B-1. (2) (7.3) (AB)-' in relation to B-1A-1. (3)

Consider the matrices A = −2 7 1

3 4 1

8 1 5 ,

B = 8 1 5

3 4 1

−2 7 1 ,

C = −2 7 1

3 4 1

2 −7 3 .

Find elementary matrices E1, E2 and E3 such that

(5.1) E1A = B,

(5.2) E1B = A,

(5.3) E2A = C,

(5.4) E3C = A.

Suppose V1,V2,VM is linearly independent in V and W€V.Prove that dim span(V1+W1,V2+W,.....VM+W)> or equal to m-1.

2.Suppose U1,U2,.....Um are finite dimensional subspaces of V.Prove that U1+U2+.....Um is finite dimensional and dim(U1+U2+.....Um)<or equal to dim U1+dim U2+.......dim Um.

Solve the following system of linear equations: x + y + z = 2 (1) 6x − 4y + 5z = 31 (2) 5x + 2y + 2z = 13 (3) The solution of the system is the ordered triple: [1] (3, − 2, 1). [2] (3, 2, 1). [3] (0, 2, 0). [4] (1, 2, 1).

Express v=(1,-2,5) in R³ as a linear combination of the vectors U₁=(1,1,1), U₂=(1,2,3), U₃=(2,-1,1)