Linear Algebra Answers

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

let

Matrix

A = 1 -1 1 B = 8 -3 -5

0 2 -1 0 1 2

2 1 3 4 -7 6

Compute A ^-1 , (B^T )^-1 and B^-1A^-1. What do you observe about

(7.1) (A ^-1 )^-1 in relation to A.

(7.2) ((B^T )^-1 ) T in relation to B^-1 .

(7.3) (AB) ^-1 in relation to B^-1 A^-1 .

Write notes on how to add, subtract and multiply matrices, and show how they may apply to solving a three system of equation , site examples on the applications of matrices to solving real world business problems