If `A=[(1,0),(3,-1),(-5,2)]` and `B=[(1,-2),(-2,2),(1,1)]` , then find the matrix 'X' such that 3A + X = 5B.

Options

A. `[(2,-10),(-19,13),(20,-1)]`
B. `[(-10,1),(13,-19),(-1,20)]`
C. `[(-10,12),(3,-19),(-1,20)]`
D. None of these

Answer

To solve the equation \(3A + X = 5B\) for the matrix \(X\), we will follow these steps:

### Step 1: Write down the matrices A and B
Given:
\[ A = \begin{pmatrix} 1 & 0 \\ 3 & -1 \\ -5 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & -2 \\ -2 & 2 \\ 1 & 1 \end{pmatrix} \]

### Step 2: Calculate \(3A\)
To find \(3A\), we multiply each element of matrix \(A\) by 3:
\[ 3A = 3 \cdot \begin{pmatrix} 1 & 0 \\ 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 \cdot 1 & 3 \cdot 0 \\ 3 \cdot 3 & 3 \cdot -1 \\ 3 \cdot -5 & 3 \cdot 2 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 9 & -3 \\ -15 & 6 \end{pmatrix} \]

### Step 3: Calculate \(5B\)
To find \(5B\), we multiply each element of matrix \(B\) by 5:
\[ 5B = 5 \cdot \begin{pmatrix} 1 & -2 \\ -2 & 2 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 5 \cdot 1 & 5 \cdot -2 \\ 5 \cdot -2 & 5 \cdot 2 \\ 5 \cdot 1 & 5 \cdot 1 \end{pmatrix} = \begin{pmatrix} 5 & -10 \\ -10 & 10 \\ 5 & 5 \end{pmatrix} \]

### Step 4: Rearrange the equation to find \(X\)
From the equation \(3A + X = 5B\), we can rearrange it to find \(X\):
\[ X = 5B - 3A \]

### Step 5: Substitute \(3A\) and \(5B\) into the equation
Now we substitute the values of \(3A\) and \(5B\):
\[ X = \begin{pmatrix} 5 & -10 \\ -10 & 10 \\ 5 & 5 \end{pmatrix} - \begin{pmatrix} 3 & 0 \\ 9 & -3 \\ -15 & 6 \end{pmatrix} \]

### Step 6: Perform the matrix subtraction
Now we subtract the corresponding elements of the matrices:
\[ X = \begin{pmatrix} 5 - 3 & -10 - 0 \\ -10 - 9 & 10 - (-3) \\ 5 - (-15) & 5 - 6 \end{pmatrix} = \begin{pmatrix} 2 & -10 \\ -19 & 13 \\ 20 & -1 \end{pmatrix} \]

### Final Result
Thus, the matrix \(X\) is:
\[ X = \begin{pmatrix} 2 & -10 \\ -19 & 13 \\ 20 & -1 \end{pmatrix} \]

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