Let \( A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} -3 & 0 & 1 \\ 2 & 4 & 0 \end{bmatrix} \). If \( A + B + C = O \), then matrix \( C \) is :
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Matrix dimensions must match for addition; both \( A \) and \( B \) are \( 2 \times 3 \), so the operation is valid.
- \( \begin{bmatrix} -3 & -3 & 5 \\ 3 & 4 & 2 \end{bmatrix} \)
- \( \begin{bmatrix} 3 & 3 & 5 -3 & -4 & -2 \end{bmatrix} \)
- \( \begin{bmatrix} 3 & 3 & -5 -3 & -4 & -2 \end{bmatrix} \)
- \( \begin{bmatrix} -3 & -3 & -5 3 & 4 & 2 \end{bmatrix} \)
The Correct Option is C
Solution and Explanation
Concept:
Matrix addition is performed by adding the corresponding elements of the matrices.
If \[ A+B+C=O, \] then: \[ C=-(A+B) \]
Step 1: Calculate \(A+B\)
Given: \[ A= \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix} \] and \[ B= \begin{bmatrix} -3 & 0 & 1 \\ 2 & 4 & 0 \end{bmatrix} \] Adding corresponding elements: \[ A+B= \begin{bmatrix} 0+(-3) & -3+0 & 4+1 \\ 1+2 & 0+4 & 2+0 \end{bmatrix} \] \[ A+B= \begin{bmatrix} -3 & -3 & 5 \\ 3 & 4 & 2 \end{bmatrix} \]
Step 2: Find matrix \(C\)
Since: \[ A+B+C=O \] Therefore: \[ C=-(A+B) \] \[ C=- \begin{bmatrix} -3 & -3 & 5 \\ 3 & 4 & 2 \end{bmatrix} \] \[ C= \begin{bmatrix} 3 & 3 & -5 \\ -3 & -4 & -2 \end{bmatrix} \]
Step 3: Compare with the options
The required matrix is: \[ C= \begin{bmatrix} 3 & 3 & -5 \\ -3 & -4 & -2 \end{bmatrix} \]
Final Answer:
\[ \boxed{ C= \begin{bmatrix} 3 & 3 & -5 \\ -3 & -4 & -2 \end{bmatrix} } \] Hence, the correct answer is Option (C).
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