Question

If \( \begin{bmatrix} 1 & 3 \\ 4 & 5 \end{bmatrix} \begin{bmatrix} x \\ 2 \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix} \), then value of \( x \) is :

• A. \( 1 \)
• B. \( 0 \)
• C. \( -1 \)
• D. \( 3 \)

Hint:

In a multiple-choice setting, you can also "plug and play." If you substitute the options into the matrix multiplication, only \( x = -1 \) satisfies both rows. For example, if \( x=1 \), row 1 would give \( 1+6=7 \), which does not equal 5. This saves time in exam conditions.

Answer 1

The Correct Option is C

Step 1: Multiply the matrix by the column.
Take each row and pair it with the column $\begin{bmatrix} x \\ 2 \end{bmatrix}$.
\[ \begin{bmatrix} 1 & 3 \\ 4 & 5 \end{bmatrix}\begin{bmatrix} x \\ 2 \end{bmatrix} = \begin{bmatrix} x + 6 \\ 4x + 10 \end{bmatrix} \]

Step 2: Match the top entries.
The result must equal $\begin{bmatrix} 5 \\ 6 \end{bmatrix}$, so the top row gives
\[ x + 6 = 5 \]

Step 3: Solve for $x$.
\[ x = 5 - 6 = -1 \]

Step 4: Quick check with the bottom row.
Put $x = -1$ into $4x + 10$. That gives $-4 + 10 = 6$, which matches. So $x = -1$ is option 3.
\[ \boxed{x = -1} \]