Question

If A =\(\begin{bmatrix}2 & 4 \\4 & 3 \end{bmatrix} \), X =\(\begin{bmatrix}n \\1 \end{bmatrix}\),B =\(\begin{bmatrix}8 \\11 \end{bmatrix}\),and AX = B, then the value of n will be:

• A. 0
• B. 1
• C. 2
• D. not defined

Answer 1

The Correct Option is C

Solution: We are solving the equation AX = B, where:

\( A = \begin{bmatrix} 2 & 4 \\ 4 & 3 \end{bmatrix}, \quad X = \begin{bmatrix} n \\ 1 \end{bmatrix}, \quad B = \begin{bmatrix} 8 \\ 11 \end{bmatrix}. \)

Substitute X into AX:

\[ \begin{bmatrix} 2 & 4 \\ 4 & 3 \end{bmatrix} \begin{bmatrix} n \\ 1 \end{bmatrix} = \begin{bmatrix} 2n + 4 \\ 4n + 3 \end{bmatrix}. \]

Equate with B:

\[ \begin{bmatrix} 2n + 4 \\ 4n + 3 \end{bmatrix} = \begin{bmatrix} 8 \\ 11 \end{bmatrix}. \]

From the first equation:

\( 2n + 4 = 8 \quad \Rightarrow \quad 2n = 4 \quad \Rightarrow \quad n = 2. \)

Verify with the second equation:

\( 4n + 3 = 11 \quad \Rightarrow \quad 4(2) + 3 = 11, \) which is true.

Thus, \( n = 2. \)