If `A = [(1,2),(9,7)]," then "A^(3)` =

Options

A. `[(163,150),(675, 613)]`
B. `[(150,675),(613, 163)]`
C. `[(163,675),(613, 150)]`
D. `[(613,675),(620, 163)]`

Answer

To find \( A^3 \) for the matrix \( A = \begin{pmatrix} 1 & 2 \\ 9 & 7 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \times A = \begin{pmatrix} 1 & 2 \\ 9 & 7 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 9 & 7 \end{pmatrix} \]

Using the matrix multiplication formula, the elements of the resulting matrix are calculated as follows:

- First row, first column:
\[ 1 \cdot 1 + 2 \cdot 9 = 1 + 18 = 19 \]

- First row, second column:
\[ 1 \cdot 2 + 2 \cdot 7 = 2 + 14 = 16 \]

- Second row, first column:
\[ 9 \cdot 1 + 7 \cdot 9 = 9 + 63 = 72 \]

- Second row, second column:
\[ 9 \cdot 2 + 7 \cdot 7 = 18 + 49 = 67 \]

Thus, we have:
\[ A^2 = \begin{pmatrix} 19 & 16 \\ 72 & 67 \end{pmatrix} \]

### Step 2: Calculate \( A^3 \)

Now, we will calculate \( A^3 \) by multiplying \( A^2 \) by \( A \):

\[ A^3 = A^2 \times A = \begin{pmatrix} 19 & 16 \\ 72 & 67 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 9 & 7 \end{pmatrix} \]

Calculating the elements of the resulting matrix:

- First row, first column:
\[ 19 \cdot 1 + 16 \cdot 9 = 19 + 144 = 163 \]

- First row, second column:
\[ 19 \cdot 2 + 16 \cdot 7 = 38 + 112 = 150 \]

- Second row, first column:
\[ 72 \cdot 1 + 67 \cdot 9 = 72 + 603 = 675 \]

- Second row, second column:
\[ 72 \cdot 2 + 67 \cdot 7 = 144 + 469 = 613 \]

Thus, we have:
\[ A^3 = \begin{pmatrix} 163 & 150 \\ 675 & 613 \end{pmatrix} \]

### Final Answer
\[ A^3 = \begin{pmatrix} 163 & 150 \\ 675 & 613 \end{pmatrix} \]