52 CHAPTER 2. LINEAR TRANSFORMATIONS

Next add (−1) times the second row to the bottom row. 1 2 2 1 0 0

0 −2 0 −1 1 0

0 0 0 −1 −1 1

At this point, you can see there will be no inverse because you have obtained a row of zerosin the left half of the augmented matrix (A|I) . Thus there will be no way to obtain I onthe left. In other words, the three systems of equations you must solve to find the inversehave no solution. In particular, there is no solution for the first column of A−1 which mustsolve

A

 x

y

z

 =

 1

0

0

because a sequence of row operations leads to the impossible equation, 0x+ 0y + 0z = −1.

2.2 Exercises

1. In 2.1 - 2.8 describe −A and 0.

2. Let A be an n×nmatrix. Show A equals the sum of a symmetric and a skew symmetricmatrix.

3. Show every skew symmetric matrix has all zeros down the main diagonal. The maindiagonal consists of every entry of the matrix which is of the form aii. It runs fromthe upper left down to the lower right.

4. Using only the properties 2.1 - 2.8 show −A is unique.

5. Using only the properties 2.1 - 2.8 show 0 is unique.

6. Using only the properties 2.1 - 2.8 show 0A = 0. Here the 0 on the left is the scalar 0and the 0 on the right is the zero for m× n matrices.

7. Using only the properties 2.1 - 2.8 and previous problems show (−1)A = −A.

8. Prove 2.17.

9. Prove that ImA = A where A is an m× n matrix.

10. Let A and be a real m × n matrix and let x ∈ Rn and y ∈ Rm. Show (Ax,y)Rm =(x,ATy

)Rn where (·, ·)Rk denotes the dot product in Rk.

11. Use the result of Problem 10 to verify directly that (AB)T= BTAT without making

any reference to subscripts.

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

13. Give an example of matrices, A,B,C such that B ̸= C, A ̸= 0, and yet AB = AC.

14. Let A =

 1 1

−2 −1

1 2

, B =

(1 −1 −2

2 1 −2

), and C =

 1 1 −3

−1 2 0

−3 −1 0

 . Find

if possible the following products. AB,BA,AC,CA,CB,BC.