Algebra questions

Question

Question 1

Matrices;

If A = 5x3 , B = 5x2 and C = 2x5 then what is A^TBC (A transpose)

 

Question 2

Cant draw the matrices below but they are 3x3

If A = 3 4 4

         1 2  2

        -2 1 3

and B = 2 1 2

             1 -4 5

             3 2 1

What is c31 ? I cant remember the exact question but it was either BC=A or AB=C im just not sure what c31 means, i know how to compute AB

 

Question 3

This is also meant to be a matrix (best attempt at drawing it)

How many free variables are there in this?

1 0 3 5 0 2 1

0 1 1 1 0 1 2

0 0 0 0 1 1 3

0 0 0 0 0 0 0

0 0 0 0 0 0 0

I thought it was a non-zero number after the leading variable 1 so i thought 10 but that wasnt right

 

Question 4

For these matrices what would the next operation be according to gauss reduction, i understand what is needed for gauss reduction these just seem to confuse me

1 -2 0 6

0 1 0 1

0 0 1 3

p is meant to stand for row

a) p1 <-- p1 - 2p3

b) p1 <-- p1 + 2p2

c) p2 <-- p2 + 1/2(p1)

d) p3 <-- p3 - 2p1

 

Another one:

1 1 4 1

0 0 1 5

0 3 3 4

a) p2 <-- p2 + p1

b) p2 <-- 1/5(p2)

c) p2 <--> p3

d) p3 <-- p3 - 2p1

 

Thank you :)

Answer 1

Question - 1)

Answer 2

Q4)

The augmented matrix is

We should operate from left to right by columns, using elementary row operations to obtain zeros in all entries directly below the leading 1 ' s.

In this case , to complete row - echelon form we need the elementary row operation.

Option b is correct .

Answer 3

Another one:

To complete the row - echelon form, we want to  turn the 0 in to a 1.

To rewrite the augmented matrix in row-echelon form, inter change p2 and p3 . 

In the next step we need the row operation .

Option C is correct .

 

Answer 4

Question2)

C31 means 3rd row and 1st column

AB = C

.

Answer 5

Q3)

Row - echelon form of the matrix is

In this case 7 free variables are including.