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

Hint:

When solving matrix equations like \( AX = B \), always ensure that you correctly multiply matrices and then solve the resulting system of equations. For a 2x2 matrix, this involves applying the distributive property of matrix multiplication. Once you obtain the system of linear equations, solve them one by one to find the unknowns. Always verify the solution by substituting it back into the original system to confirm its accuracy.

Answer 1

The Correct Option is C

Given the equation \( AX = B \), where: 
A = \(\begin{bmatrix}2 & 4 \\ 4 & 3 \end{bmatrix} \),
X = \(\begin{bmatrix}n \\ 1 \end{bmatrix} \),
B = \(\begin{bmatrix}8 \\ 11 \end{bmatrix} \).
We need to find the value of \( n \).

First, perform the matrix multiplication for \( AX \):
\(\begin{bmatrix}2 & 4 \\ 4 & 3 \end{bmatrix} \begin{bmatrix}n \\ 1 \end{bmatrix} = \begin{bmatrix}2n + 4 \\ 4n + 3 \end{bmatrix} \).

Set it equal to matrix \( B \):
\(\begin{bmatrix}2n + 4 \\ 4n + 3 \end{bmatrix} = \begin{bmatrix}8 \\ 11 \end{bmatrix} \).

Equating the corresponding components:
\(2n + 4 = 8\) and \(4n + 3 = 11\).

Solve the first equation:
\(2n + 4 = 8\)
Subtract 4 from both sides:
\(2n = 4\)
Divide by 2:
\(n = 2\).

Check the second equation to confirm:
\(4n + 3 = 11\)
Substitute \(n = 2\):
\(4(2) + 3 = 11\)
\(8 + 3 = 11\)
Which is true. Thus, \(n = 2\) satisfies both equations.

Therefore, the value of \( n \) is 2.

Answer 2

We are solving the equation \( AX = B \), where:

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

Step 1: Substitute \( X \) into \( AX \):

Multiply matrix \( A \) with vector \( X \): \[ \begin{bmatrix} 2 & 4 \\ 4 & 3 \end{bmatrix} \begin{bmatrix} n \\ 1 \end{bmatrix} = \begin{bmatrix} 2n + 4 \\ 4n + 3 \end{bmatrix}. \]

Step 2: Equate with \( B \):

Now, equate the result of \( AX \) to matrix \( B \): \[ \begin{bmatrix} 2n + 4 \\ 4n + 3 \end{bmatrix} = \begin{bmatrix} 8 \\ 11 \end{bmatrix}. \]

Step 3: Solve the first equation:

From the first row of the matrix, we have the equation: \[ 2n + 4 = 8 \quad \Rightarrow \quad 2n = 4 \quad \Rightarrow \quad n = 2. \]

Step 4: Verify with the second equation:

From the second row of the matrix, we have the equation: \[ 4n + 3 = 11 \quad \Rightarrow \quad 4(2) + 3 = 11, \] which simplifies to: \[ 8 + 3 = 11, \] which is true.

Conclusion: Thus, \( n = 2 \).