5.3. EXERCISES 101

3. Here are some matrices:

A =

 1 23 21 −1

 ,B =

(2 −5 2−3 2 1

),

C =

(1 25 0

),D =

(−1 14 −3

),E =

(13

).

Find if possible −3AT ,3B− AT ,AC,CA,AE,ET B,BE,DE,EET ,ET E. If it is notpossible explain why.

4. Here are some matrices:

A =

 1 23 21 −1

 ,B =

(2 −5 2−3 2 1

),

C =

(1 25 0

),D =

(−14

),E =

(13

).

Find the following if possible and explain why it is not possible if this is the case.

AD,DA,DT B,DT BE,ET D,DET .

5. Let A =

 1 1−2 −11 2

, B =

(1 −1 −22 1 −2

), and C =

 1 1 −3−1 2 0−3 −1 0

 .

Find if possible.

(a) AB

(b) BA

(c) AC

(d) CA

(e) CB

(f) BC

6. Suppose A and B are square matrices of the same size. Which of the following arecorrect?

(a) (A−B)2 = A2−2AB+B2

(b) (AB)2 = A2B2

(c) (A+B)2 = A2 +2AB+B2

(d) (A+B)2 = A2 +AB+BA+B2

(e) A2B2 = A(AB)B

(f) (A+B)3 = A3 +3A2B+3AB2 +B3