Linear Algebra Answers

Which of the following is a linear equation in x; y and z? 1. x 2 − e 2y = 3z, where e = 2.71828 .... 2. 2π ln(e z ) − 2y + z = ln(3) − x. 3. p y −1 + 4y − 2z = 7x. 4. y + 4y − 2z = 7x 3 .

which of the following is a linear equation in x;y and z?

1. x-e^2y=3z, where e=2.71828....

2. 2πIn(e 1/z)-2y+z=In(3)-x

3. √y^-2 +4y-2z=7x

4. y+4y-2z=7x^3

Show that if A is a matrix with a row of zeros (or a column of zeros), then A cannot have an inverse

"Cosy dy\/dx" = "Sin^2x Cosx"

Find det(C) if

C = λ λ + 1

λ λ − 1

consider the linear transformation defined by F ( x , y , z ) = ( y z , x 2 ) . find f ( 2 , 3 , 4 )

Consider the matrices

A = 3 0 2 , B = -5 1 1 , C = 1 1 1

4 -6 3 0 3 0 2 3 -1

-2 1 8 7 6 2 3 -5 -7

Verify the following expressions (where possible and give reasons)

(i) A + (B + C) = (A + B) + C and A(BC) = (AB)C.

(ii) (a − b)C = aC + bC and a(B − C) = aB − aC, where a = −2, b = 3 .

(iii) (A^T)^T = A and (A − B)^T = A ^T− B^T

Consider the matrix

A = 1 4

2 3

(a) Compute A^-1

(b) Find det(A^-1)

(c) Deduce a relation (if it exists) between det(A) and det(A^-1)

Consider the given matrix

B = 2 2 0

1 0 1

0 1 1

Find detB and use it to determine whether or not B is invertible, and if so, find B^-1 . (Hint: Use the matrix equation BX = I)

Show that if A is an n × n matrix, then AA^T and A + A^T are symmetric