QuestionSection 9.2 Follow-up Exercises Determine the dimension of each of the following matrices and find the transpose. $1\left(\begin{array}{lll}8 & -8 & 5\end{array}\right.$ 3) $3\left(\begin{array}{rr}0 & 1 \\ 5 & 2 \\ -6 & 8 \\ -2 & 4\end{array}\right)$ $\begin{array}{l} \mathbf{2}\left(\begin{array}{rr} 2 & 6 \\ -3 & 8 \end{array}\right) \\ \mathbf{4}\left(\begin{array}{rrr} 2 & 10 & -1 \\ -3 & -5 & 0 \\ 4 & -8 & 2 \end{array}\right) \\ \mathbf{6}\left(\begin{array}{rrrr} -6 & 3 & 2 & 4 \\ 2 & 3 & 3 & 4 \\ 2 & -1 & 5 & 8 \end{array}\right) \\ \mathbf{8}\left(\begin{array}{rrrrr} 1 & 3 & 5 & 7 & 9 \\ 2 & 4 & 6 & 8 & 10 \end{array}\right) \\ \mathbf{1 0}\left(\begin{array}{rrrrr} 6 & 1 & 2 & 3 & 5 \\ 2 & 0 & 4 & 6 & 1 \\ 3 & 1 & -2 & 3 & 5 \\ 4 & 3 & 2 & 1 & 0 \end{array}\right) \end{array}$ $\begin{array}{l} \mathbf{5}\left(\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right) \\ \mathbf{7}\left(\begin{array}{l} 1 \\ 2 \\ 3 \\ 4 \end{array}\right) \\ \mathbf{9}\left(\begin{array}{lll} 1 & 3 & 5 \\ 6 & 4 & 2 \\ 0 & 1 & 2 \\ 4 & 6 & 3 \\ 5 & 1 & 2 \end{array}\right) \end{array}$

Studdy Solution

STEP 1

1. The dimension of a matrix is given by the number of rows by the number of columns.
2. The transpose of a matrix is obtained by swapping its rows with columns.

STEP 2

1. Determine the dimension of each matrix.
2. Find the transpose of each matrix.

STEP 3

Determine the dimension of each matrix.
1. Matrix $1$ : $\begin{pmatrix} 8 & -8 & 5 \end{pmatrix}$ - Dimension: $1 \times 3$
2. Matrix $3$ : $\begin{pmatrix} 0 & 1 \\ 5 & 2 \\ -6 & 8 \\ -2 & 4 \end{pmatrix}$ - Dimension: $4 \times 2$
3. Matrix $2$ : $\begin{pmatrix} 2 & 6 \\ -3 & 8 \end{pmatrix}$ - Dimension: $2 \times 2$
4. Matrix $4$ : $\begin{pmatrix} 2 & 10 & -1 \\ -3 & -5 & 0 \\ 4 & -8 & 2 \end{pmatrix}$ - Dimension: $3 \times 3$
5. Matrix $6$ : $\begin{pmatrix} -6 & 3 & 2 & 4 \\ 2 & 3 & 3 & 4 \\ 2 & -1 & 5 & 8 \end{pmatrix}$ - Dimension: $3 \times 4$
6. Matrix $8$ : $\begin{pmatrix} 1 & 3 & 5 & 7 & 9 \\ 2 & 4 & 6 & 8 & 10 \end{pmatrix}$ - Dimension: $2 \times 5$
7. Matrix $10$ : $\begin{pmatrix} 6 & 1 & 2 & 3 & 5 \\ 2 & 0 & 4 & 6 & 1 \\ 3 & 1 & -2 & 3 & 5 \\ 4 & 3 & 2 & 1 & 0 \end{pmatrix}$ - Dimension: $4 \times 5$
8. Matrix $5$ : $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ - Dimension: $3 \times 3$
9. Matrix $7$ : $\begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix}$ - Dimension: $4 \times 1$
10. Matrix $9$ : $\begin{pmatrix} 1 & 3 & 5 \\ 6 & 4 & 2 \\ 0 & 1 & 2 \\ 4 & 6 & 3 \\ 5 & 1 & 2 \end{pmatrix}$ - Dimension: $5 \times 3$

STEP 4

Find the transpose of each matrix.
1. Transpose of Matrix $1$ : $ \begin{pmatrix} 8 \\ -8 \\ 5 \end{pmatrix} \]
2. Transpose of Matrix $3$ : $ \begin{pmatrix} 0 & 5 & -6 & -2 \\ 1 & 2 & 8 & 4 \end{pmatrix} \]
3. Transpose of Matrix $2$ : $ \begin{pmatrix} 2 & -3 \\ 6 & 8 \end{pmatrix} \]
4. Transpose of Matrix $4$ : $ \begin{pmatrix} 2 & -3 & 4 \\ 10 & -5 & -8 \\ -1 & 0 & 2 \end{pmatrix} \]
5. Transpose of Matrix $6$ : $ \begin{pmatrix} -6 & 2 & 2 \\ 3 & 3 & -1 \\ 2 & 3 & 5 \\ 4 & 4 & 8 \end{pmatrix} \]
6. Transpose of Matrix $8$ : $ \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ 7 & 8 \\ 9 & 10 \end{pmatrix} \]
7. Transpose of Matrix $10$ : $ \begin{pmatrix} 6 & 2 & 3 & 4 \\ 1 & 0 & 1 & 3 \\ 2 & 4 & -2 & 2 \\ 3 & 6 & 3 & 1 \\ 5 & 1 & 5 & 0 \end{pmatrix} \]
8. Transpose of Matrix $5$ : $ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
9. Transpose of Matrix $7$ : $ \begin{pmatrix} 1 & 2 & 3 & 4 \end{pmatrix} \]
10. Transpose of Matrix $9$ : $ \begin{pmatrix} 1 & 6 & 0 & 4 & 5 \\ 3 & 4 & 1 & 6 & 1 \\ 5 & 2 & 2 & 3 & 2 \end{pmatrix} \]

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