Problems

Problem 1

For this problem let

\[A = \begin{bmatrix} 3 & 1 & -2 \\ 5 & -4 & 3 \end{bmatrix}, B = \begin{bmatrix} 8 & 2\\ -6 & -3\\ 2 & -4 \end{bmatrix}\]

\[C = \begin{bmatrix} 2 & 3 \\ -3 & 2 \end{bmatrix}, D = \begin{bmatrix} 4 & 6 \\ -2 & 5 \end{bmatrix}, E = \begin{bmatrix} -6 \\ 4 \end{bmatrix}\]

Compute the following. If the operation is not possible explain why

1. \(-3A+B\)
2. \(B - 3A^T\)
3. \(I_3-AB\)
4. \(AB-2I_2\)
5. \(AC\)
6. \(DA\)
7. \(E^TCE\)
8. \(BD+A^T\)

Problem 2

Let \(A = \begin{bmatrix} \uparrow & \uparrow & \uparrow \\ \mathbf{c}_1 & \mathbf{c}_2 & \mathbf{c}_3 \\ \downarrow & \downarrow & \downarrow \end{bmatrix} = \begin{bmatrix} \leftarrow \mathbf{r}_1 \rightarrow \\ \leftarrow \mathbf{r}_2 \rightarrow \\ \leftarrow \mathbf{r}_3 \rightarrow \\ \leftarrow \mathbf{r}_4 \rightarrow \end{bmatrix}\)

and suppose that \(A \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ 2 \\ -9 \end{bmatrix}\). For the following write the vectors in terms of the columns or rows of \(A\).

1. What are the dimensions of \(A\)?

2. Is any row of \(A\) orthogonal to \((1,1,-1)\)? (Actually, the question is if the transpose of any row is orthogonal to \((1,1,-1)\))

3. Write a vector \(\mathbf{v}\) in terms of the rows of \(A\) that satisfies \(\mathbf{v} \cdot (1,1,-1) = 3\)

4. Write a vector \(\mathbf{v}\) in terms of the rows of \(A\) that satisfies \(\mathbf{v} \cdot (1,1,-1) = 1\)

Problem 3

Let \(A = \begin{bmatrix} \uparrow & \uparrow & \uparrow \\ \mathbf{c}_1 & \mathbf{c}_2 & \mathbf{c}_3 \\ \downarrow & \downarrow & \downarrow \end{bmatrix}\), \(B = \begin{bmatrix} \uparrow & \uparrow & \uparrow \\ \mathbf{d}_1 & \mathbf{d}_2 & \mathbf{d}_3 \\ \downarrow & \downarrow & \downarrow \end{bmatrix}\) and suppose that \(A^tB = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\)

1. What are the dimensions of \(A\) and \(B\)?

2. Find the following dot products: \(\mathbf{c}_1 \cdot \mathbf{d}_1\), \(\mathbf{c}_3 \cdot \mathbf{d}_1\), \(\mathbf{c}_2 \cdot \mathbf{d}_3\)

3. Find a constant \(\alpha \in \mathbb{R}\) such that \(\mathbf{c}_1 - \alpha \mathbf{c}_2\) is orthogonal to \(\mathbf{d}_3\)

Problem 4

Let \(A = \begin{bmatrix} \uparrow & \uparrow & \uparrow \\ \mathbf{c}_1 & \mathbf{c}_2 & \mathbf{c}_3 \\ \downarrow & \downarrow & \downarrow \end{bmatrix}\) and assume that \(A^tA = \begin{bmatrix} 1 & -1 & 1 \\ -1 & 2 & 0 \\ 1 & 0 & 3 \end{bmatrix}\)

1. Find the following dot products: \(\mathbf{c}_1 \cdot \mathbf{c}_2\), \(\mathbf{c}_1 \cdot \mathbf{c}_3\), \(\mathbf{c}_2 \cdot \mathbf{c}_3\)

2. Find \(\|\mathbf{c}_1\|\), \(\|\mathbf{c}_2\|\), \(\|\mathbf{c}_3\|\)

3. Find \(\|\mathbf{c}_1 + 3\mathbf{c}_2\|\) and \(\|3\mathbf{c}_2 - 4\mathbf{c}_3\|\)

---

For the following problems we have:

\(A = \begin{bmatrix}\uparrow&\uparrow&\uparrow\\ \mathbf{c}_1&\mathbf{c}_2 &\mathbf{c}_3\\ \downarrow&\downarrow&\downarrow\end{bmatrix} = \begin{bmatrix} \leftarrow \mathbf{r}_1 \rightarrow \\ \leftarrow \mathbf{r}_2 \rightarrow \\ \leftarrow \mathbf{r}_3 \rightarrow \\ \leftarrow \mathbf{r}_4 \rightarrow \end{bmatrix}\)

\(E_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\),   \(E_2 = \begin{bmatrix} 1 & 0 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\),  

\(E_3 = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & -1 \end{bmatrix}\),   \(E_4 = \begin{bmatrix} 1 & 0 & 1 & -10 \\ 2 & 2 & 1 & 0 \\ 0 & -2 & -1 & 0 \\ 4 & 0 & 0 & -1 \end{bmatrix}\)

Problem 5

For each of the following problems, indicate if the matrices can be multiplied. If they can, express the answer in terms of the rows of \(A\) or the columns of \(A\). If they cannot be multiplied, explain why.

1. \(AE_1\)

2. \(E_1A\)

3. \(AE_2\)

4. \(E_2A\)

5. \(AE_3\)

6. \(E_3A\)

7. \(AE_4\)

8. \(E_4A\)

Problem 6

Suppose that \(AB_1 = \begin{bmatrix} \uparrow & \uparrow & \uparrow & \uparrow \\ (\mathbf{c}_1-3\mathbf{c}_2) & \mathbf{c}_3 & \mathbf{c}_2 & (8\mathbf{c}_2-\mathbf{c}_3) \\ \downarrow & \downarrow & \downarrow & \downarrow \end{bmatrix}\). Find \(B_1\)

Problem 7

Suppose that \(B_2A = \begin{bmatrix} \leftarrow \mathbf{r}_1 \rightarrow \\ \leftarrow \mathbf{r}_2 \rightarrow \\ \leftarrow \mathbf{r}_3 \rightarrow \\ \leftarrow \mathbf{r}_4 \rightarrow \end{bmatrix}\). Find \(B_2\)

Problem 8

Suppose that \(B_3A = \begin{bmatrix} \leftarrow (\mathbf{r}_1+\mathbf{r}_2-\mathbf{r}_3) \rightarrow \\ \leftarrow (\mathbf{r}_2-\mathbf{r}_3+\mathbf{r}_4) \rightarrow \end{bmatrix}\). Find \(B_3\)

Problem 9

Suppose that \(B_4E_1 = A\). Find \(B_4\)

Problem 10 (Optional)

Suppose that \(B_5\begin{bmatrix} \leftarrow (\mathbf{r}_1+\mathbf{r}_2+\mathbf{r}_3+\mathbf{r}_4) \rightarrow \\ \leftarrow (\mathbf{r}_2+\mathbf{r}_3+\mathbf{r}_4) \rightarrow \\ \leftarrow (\mathbf{r}_3+\mathbf{r}_4) \rightarrow \\ \leftarrow \mathbf{r}_4 \rightarrow \end{bmatrix} = A\). Find \(B_5\)