Let `A={:[(2,3,5),(1,0,2),(3,4,5)]:}andA+B-4I=0`, then B is equal to

Options

A. `{:[(2,-3,-5),(-1,4,-2),(-3,-4,-1)]:}`
B. `{:[(2,3,5),(1,-4,2),(3,4,1)]:}`
C. `{:[(2,-3,-5),(-1,4,-2),(-3,-4,-1)]:}`
D. `{:[(2,3,5),(-1,4,-2),(3,4,1)]:}`

Answer

To solve the problem, we need to find the matrix \( B \) given the equation \( A + B - 4I = 0 \), where \( A \) is a matrix and \( I \) is the identity matrix.

### Step-by-Step Solution:

1. **Identify the Matrix \( A \)**:
Given \( A = \begin{pmatrix} 2 & 3 & 5 \\ 1 & 0 & 2 \\ 3 & 4 & 5 \end{pmatrix} \).

2. **Write Down the Identity Matrix \( I \)**:
The identity matrix \( I \) of order 3 is:
\[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

3. **Multiply the Identity Matrix by 4**:
We need to calculate \( 4I \):
\[ 4I = 4 \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]

4. **Set Up the Equation**:
From the equation \( A + B - 4I = 0 \), we can rearrange it to find \( B \):
\[ B = 4I - A \]

5. **Calculate \( B \)**:
Now substitute the values of \( A \) and \( 4I \):
\[ B = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{pmatrix} - \begin{pmatrix} 2 & 3 & 5 \\ 1 & 0 & 2 \\ 3 & 4 & 5 \end{pmatrix} \]
Perform the subtraction element-wise:
\[ B = \begin{pmatrix} 4-2 & 0-3 & 0-5 \\ 0-1 & 4-0 & 0-2 \\ 0-3 & 0-4 & 4-5 \end{pmatrix} \]
Simplifying this gives:
\[ B = \begin{pmatrix} 2 & -3 & -5 \\ -1 & 4 & -2 \\ -3 & -4 & -1 \end{pmatrix} \]

6. **Final Result**:
Thus, the matrix \( B \) is:
\[ B = \begin{pmatrix} 2 & -3 & -5 \\ -1 & 4 & -2 \\ -3 & -4 & -1 \end{pmatrix} \]