1 pt Let A be a 2x 6 matrix What must aland be f we define

(1 pt) Let A be a 2x 6 matrix. What must aland & be f we define the linear transformation by T : R\" R as T z)-Az

Solution

-1

3

1

0

1

4

0

1

We will reduce A to its RREF as under:

Multiply the 1st row by -1

              Add -1 times the 1st row to the 2nd row

              Multiply the 2nd row by 1/7

              Add 3 times the 2nd row to the 1st row

Then the RREF of A is

1

0

-4/7

3/7

0

1

1/7

1/7

Therefore, e₁ = -4/7(-1,1)^T + 1/7(3,4)^T   and e₂ = 3/7(-1,1)^T +1/7(3,4)^T. Now, since T is a linear transformation, we have T( e₁)= -4/7T(-1,1)^T +1/7(3,4)^T=-4/7(0,5)^T+1/7(14,-8)^T=(0,-20/7)^T+  (2,-8/7)^T= ( 2,-4)^T andT(e₂ )= 3/7 T(-1,1)^T +1/7T(3,4)^T = 3/7 (0,5)^T +1/7(14,-8)^T= (0,15/7)^T + (2,-8/7)^T = (2,1)^T. Then,TX = MX, where M = [ T(e₁ ),T(e₂)] =

2

2

-4

1

4. Let the given matrices be denoted by A,B and C. Then AB =

-10

15

-6

6

0

0

Then AB +C =

-9

17

-3

10

5

6

-1	3	1	0
1	4	0	1