Question

Which of the following elementary operations has been applied to the matrix $A = \begin{bmatrix} 8 & 5 \\ 2 & 8 \end{bmatrix}$ such that the new matrix is $\begin{bmatrix} 12 & 21 \\ 2 & 8 \end{bmatrix}$?

• A. $R_1 \rightarrow R_1 - 2R_2$
• B. $R_1 \rightarrow 2R_1 + R_2$
• C. $R_1 \rightarrow R_1 + R_2$
• D. $R_1 \rightarrow R_1 + 2R_2$

Answer 1

Answer: (D)

D

Answer 2

The original matrix is $A = \begin{bmatrix} 8 & 5 \\ 2 & 8 \end{bmatrix}$. The new matrix is $\begin{bmatrix} 12 & 21 \\ 2 & 8 \end{bmatrix}$. The second row is unchanged. We check the options for operations on the first row. Applying $R_1 \rightarrow R_1 + 2R_2$ to the first row $\begin{bmatrix} 8 & 5 \end{bmatrix}$ gives $\begin{bmatrix} 8 & 5 \end{bmatrix} + 2\begin{bmatrix} 2 & 8 \end{bmatrix} = \begin{bmatrix} 8 & 5 \end{bmatrix} + \begin{bmatrix} 4 & 16 \end{bmatrix} = \begin{bmatrix} 12 & 21 \end{bmatrix}$. This matches the first row of the new matrix.