Question

Let \( A = \begin{bmatrix} 5 & 0 \\ 1 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \). If \( 4A + 5B - C = 0 \), then the matrix \( C \) is:

• A. \( \begin{bmatrix} 5 & 25 \\ -1 & 0 \end{bmatrix} \)
• B. \( \begin{bmatrix} 20 & 5 \\ -1 & 0 \end{bmatrix} \)
• C. \( \begin{bmatrix} 5 & -1 \\ 0 & 25 \end{bmatrix} \)
• D. \( \begin{bmatrix} 5 & 25 \\ -1 & 5 \end{bmatrix} \)
• E. \( \begin{bmatrix} 0 & 5 \\ 5 & 25 \end{bmatrix} \)

Hint:

Always rearrange the matrix equation to isolate the unknown matrix (like \( C \)) on one side before plugging in the numerical values. This prevents sign errors during the arithmetic phase.

Answer 1

The Correct Option is B

Concept: Matrix addition and scalar multiplication follow distributive and commutative laws similar to basic algebra. In the equation \( 4A + 5B - C = 0 \), we can isolate \( C \) by rearranging the terms: \( C = 4A + 5B \). Calculations must be performed element-wise.

Step 1: Calculate the scaled matrices \( 4A \) and \( 5B \).
First, multiply every element in matrix \( A \) by 4: \[ 4A = 4 \times \begin{bmatrix} 5 & 0 1 & 0 \end{bmatrix} = \begin{bmatrix} 20 & 0 \\ 4 & 0 \end{bmatrix} \] Next, multiply every element in matrix \( B \) by 5: \[ 5B = 5 \times \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 5 \\ -5 & 0 \end{bmatrix} \]

Step 2: Add the resulting matrices to determine matrix \( C \).
Matrix addition requires adding elements at the same position: \[ C = \begin{bmatrix} 20 + 0 & 0 + 5 \\ 4 + (-5) & 0 + 0 \end{bmatrix} \] Simplifying each element: \[ C = \begin{bmatrix} 20 & 5 \\ -1 & 0 \end{bmatrix} \]