## 1.7 Problems

1. Build a $$4 \times 3$$ matrix with the numbers 1 through 3 in each column. Try the same with the numbers 1 through 4 in each row.

2. Extract the elements in the 1st and 2nd rows and 1st and 2nd columns (you’ll have a $$2 \times 2$$ matrix). Show the R code that will do this.

3. Build a $$4 \times 3$$ matrix with the numbers 1 through 12 by row (meaning the first row will have the numbers 1 through 3 in it).

4. Extract the 3rd row of the above. Show R code to do this where you end up with a vector and how to do this where you end up with a $$1 \times 3$$ matrix.

5. Build a $$4 \times 3$$ matrix that is all 1s except a 2 in the (2,3) element (2nd row, 3rd column).

6. Take the transpose of the above.

7. Build a $$4 \times 4$$ diagonal matrix with 1 through 4 on the diagonal.

8. Build a $$5 \times 5$$ identity matrix.

9. Replace the diagonal in the above matrix with 2 (the number 2).

10. Build a matrix with 2 on the diagonal and 1s on the offdiagonals.

11. Take the inverse of the above.

12. Build a $$3 \times 3$$ matrix with the first 9 letters of the alphabet. First column should be “a,” “b,” “c.” letters[1:9] gives you these letters.

13. Replace the diagonal of this matrix with the word “cat.”

14. Build a $$4 \times 3$$ matrix with all 1s. Multiply by a $$3 \times 4$$ matrix with all 2s.

15. If $$\mathbf{A}$$ is a $$4 \times 3$$ matrix, is $$\mathbf{A} \mathbf{A}$$ possible? Is $$\mathbf{A} \mathbf{A}^\top$$ possible? Show how to write $$\mathbf{A}\mathbf{A}^\top$$ in R.

16. In the equation, $$\mathbf{A} \mathbf{B} = \mathbf{C}$$, let $$\mathbf{A}=\left[ \begin{smallmatrix}1&4&7\\2&5&8\\3&6&9\end{smallmatrix}\right]$$. Build a $$3 \times 3$$ $$\mathbf{B}$$ matrix with only 1s and 0s such that the values on the diagonal of $$\mathbf{C}$$ are 1, 8, 6 (in that order). Show your R code for $$\mathbf{A}$$, $$\mathbf{B}$$ and $$\mathbf{A} \mathbf{B}$$.

17. Same $$\mathbf{A}$$ matrix as above and same equation $$\mathbf{A} \mathbf{B} = \mathbf{C}$$. Build a $$3 \times 3$$ $$\mathbf{B}$$ matrix such that $$\mathbf{C}=2\mathbf{A}$$. So $$\mathbf{C}=\left[ \begin{smallmatrix}2&8&14\\ 4&10&16\\ 6&12&18\end{smallmatrix}\right]$$. Hint, $$\mathbf{B}$$ is diagonal.

18. Same $$\mathbf{A}$$ and $$\mathbf{A} \mathbf{B}=\mathbf{C}$$ equation. Build a $$\mathbf{B}$$ matrix to compute the row sums of $$\mathbf{A}$$. So the first row sum’ would be $$1+4+7$$, the sum of all elements in row 1 of $$\mathbf{A}$$. $$\mathbf{C}$$ will be $$\left[ \begin{smallmatrix}12\\ 15\\ 18\end{smallmatrix}\right]$$, the row sums of $$\mathbf{A}$$. Hint, $$\mathbf{B}$$ is a column matrix (1 column).

19. Same $$\mathbf{A}$$ matrix as above but now equation $$\mathbf{B} \mathbf{A} = \mathbf{C}$$. Build a $$\mathbf{B}$$ matrix to compute the column sums of $$\mathbf{A}$$. So the first column sum’ would be $$1+2+3$$. $$\mathbf{C}$$ will be a $$1 \times 3$$ matrix.

20. Let $$\mathbf{A} \mathbf{B}=\mathbf{C}$$ equation but $$\mathbf{A}=\left[ \begin{smallmatrix}2&1&1\\1&2&1\\1&1&2\end{smallmatrix}\right]$$ (so A=diag(3)+1). Build a $$\mathbf{B}$$ matrix such that $$\mathbf{C}=\left[ \begin{smallmatrix}3\\ 3\\ 3\end{smallmatrix}\right]$$. Hint, you need to use the inverse of $$\mathbf{A}$$.