Question

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 :

• A. \( \begin{bmatrix} -3 & -3 & 5\\ 3 & 4 & 2 \end{bmatrix} \)
• B. \( \begin{bmatrix} 3 & 3 & 5 \\-3 & -4 & -2 \end{bmatrix} \)
• C. \( \begin{bmatrix} 3 & 3 & -5 \\-3 & -4 & -2 \end{bmatrix} \)
• D. \( \begin{bmatrix} -3 & -3 & -5 \\ 3 & 4 & 2 \end{bmatrix} \)

Hint:

To avoid sign mistakes when solving algebra with matrices like \( A+B+C=O \), compute the sum element-by-element first, then flip the sign of every single component to reveal \( C \).

Answer 1

The Correct Option is C

Concept:
Matrix addition is performed by adding the corresponding entries of matrices having the same order. The zero matrix \(O\) is a matrix in which every element is \(0\). Given \[ A+B+C=O, \] we can determine matrix \(C\) using \[ C=-(A+B). \] 

Step 1: Compute the sum matrix \(A+B\). Add the corresponding entries of matrices \(A\) and \(B\): \[ A+B= \begin{bmatrix} 0 & -3 & 4\\ 1 & 0 & 2 \end{bmatrix} + \begin{bmatrix} -3 & 0 & 1\\ 2 & 4 & 0 \end{bmatrix}. \] Adding the corresponding elements, \[ A+B= \begin{bmatrix} 0+(-3) & -3+0 & 4+1\\ 1+2 & 0+4 & 2+0 \end{bmatrix}. \] Hence, \[ A+B= \begin{bmatrix} -3 & -3 & 5\\ 3 & 4 & 2 \end{bmatrix}. \] 

Step 2: Find matrix \(C\). From the given condition, \[ A+B+C=O, \] we obtain \[ C=O-(A+B)=-(A+B). \] Therefore, \[ C= -\begin{bmatrix} -3 & -3 & 5\\ 3 & 4 & 2 \end{bmatrix}. \] Multiplying each entry by \(-1\), \[ C= \begin{bmatrix} 3 & 3 & -5\\ -3 & -4 & -2 \end{bmatrix}. \] 
Final Answer: \[ \boxed{ C= \begin{bmatrix} 3 & 3 & -5\\ -3 & -4 & -2 \end{bmatrix} } \]