18.3. EXERCISES 393

2. 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 −3A,3B−A,AC,CA,AE,EA,BE,DE. If it is not possible explainwhy.

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. 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

(g) (A+B)(A−B) = A2 −B2

6. Let A =

(−1 −13 3

). Find all 2×2 matrices, B such that AB = 0.

7. Let x= (−1,−1,1) and y = (0,1,2) . Find xTy and xyT if possible.