If `A={:[(1,2,3),(-2,5,7)]:}and2A-3B={:[(4,5,-9),(1,2,3)]:}` then B is equal to

Options

A. `(1)/(3){:[(-2,-1,15),(5,8,-11)]:}`
B. `(1)/(3){:[(2,1,-15),(5,-8,-11)]:}`
C. `(1)/(3){:[(2,-1,15),(5,8,11)]:}`
D. `(1)/(3){:[(-2,-1,15),(-5,8,11)]:}`

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step-by-Step Solution:

1. **Identify the given matrices:**
We are given:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ -2 & 5 & 7 \end{pmatrix} \]
and
\[ 2A - 3B = \begin{pmatrix} 4 & 5 & -9 \\ 1 & 2 & 3 \end{pmatrix} \]

2. **Calculate \(2A\):**
To find \(2A\), we multiply each element of matrix \(A\) by 2:
\[ 2A = 2 \times \begin{pmatrix} 1 & 2 & 3 \\ -2 & 5 & 7 \end{pmatrix} = \begin{pmatrix} 2 \times 1 & 2 \times 2 & 2 \times 3 \\ 2 \times -2 & 2 \times 5 & 2 \times 7 \end{pmatrix} = \begin{pmatrix} 2 & 4 & 6 \\ -4 & 10 & 14 \end{pmatrix} \]

3. **Set up the equation:**
We now substitute \(2A\) into the equation \(2A - 3B = \begin{pmatrix} 4 & 5 & -9 \\ 1 & 2 & 3 \end{pmatrix}\):
\[ \begin{pmatrix} 2 & 4 & 6 \\ -4 & 10 & 14 \end{pmatrix} - 3B = \begin{pmatrix} 4 & 5 & -9 \\ 1 & 2 & 3 \end{pmatrix} \]

4. **Rearranging the equation:**
To isolate \(3B\), we add \(3B\) to both sides and subtract the matrix on the right from the left:
\[ 2A - \begin{pmatrix} 4 & 5 & -9 \\ 1 & 2 & 3 \end{pmatrix} = 3B \]

5. **Perform the subtraction:**
We perform the subtraction element-wise:
\[ \begin{pmatrix} 2 - 4 & 4 - 5 & 6 - (-9) \\ -4 - 1 & 10 - 2 & 14 - 3 \end{pmatrix} = \begin{pmatrix} -2 & -1 & 15 \\ -5 & 8 & 11 \end{pmatrix} \]

6. **Express \(B\):**
Now we have:
\[ 3B = \begin{pmatrix} -2 & -1 & 15 \\ -5 & 8 & 11 \end{pmatrix} \]
To find \(B\), we divide each element of the matrix by 3:
\[ B = \frac{1}{3} \begin{pmatrix} -2 & -1 & 15 \\ -5 & 8 & 11 \end{pmatrix} = \begin{pmatrix} -\frac{2}{3} & -\frac{1}{3} & 5 \\ -\frac{5}{3} & \frac{8}{3} & \frac{11}{3} \end{pmatrix} \]

### Final Result:
Thus, the matrix \(B\) is:
\[ B = \begin{pmatrix} -\frac{2}{3} & -\frac{1}{3} & 5 \\ -\frac{5}{3} & \frac{8}{3} & \frac{11}{3} \end{pmatrix} \]