Question

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

• A. -1
• B. 0
• C. 1
• D. 3

Answer 1

The Correct Option is A

To solve the problem, we need to determine the value of \( x \) in the matrix equation: \[\begin{bmatrix} 1 & 3 \\ 4 & 5 \end{bmatrix} \begin{bmatrix} x \\ 2 \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}\]

First, carry out the matrix multiplication on the left side:

\[\begin{bmatrix} 1 & 3 \\ 4 & 5 \end{bmatrix} \begin{bmatrix} x \\ 2 \end{bmatrix} = \begin{bmatrix} (1 \cdot x) + (3 \cdot 2) \\ (4 \cdot x) + (5 \cdot 2) \end{bmatrix} = \begin{bmatrix} x + 6 \\ 4x + 10 \end{bmatrix}\]

This results in the equation:

\[\begin{bmatrix} x + 6 \\ 4x + 10 \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}\]

From this, we have two separate equations:

1. \(x + 6 = 5\)

2. \(4x + 10 = 6\)

Solve the first equation:

\[x + 6 = 5 \]

Subtract 6 from both sides:

\[x = 5 - 6\]

\[x = -1\]

Check the solution with the second equation:

\[4x + 10 = 6\]

Substitute \(x = -1\):

\[4(-1) + 10 = 6\]

\[-4 + 10 = 6\]

\[6 = 6\]

The solution is consistent with both equations. Therefore, the value of \(x\) is \(-1\), which is the correct answer.

Answer 2

To find the value of \( x \) in the given matrix equation, we start by considering the matrices and their multiplication.
The equation is:
\[ \begin{bmatrix} 1 & 3 \\ 4 & 5 \end{bmatrix} \begin{bmatrix} x \\ 2 \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix} \]

Perform matrix multiplication on the left-hand side:
\[ \begin{bmatrix} 1 \times x + 3 \times 2 \\ 4 \times x + 5 \times 2 \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix} \]

This results in the system of equations:
\( 1x + 6 = 5 \) (Equation 1)
\( 4x + 10 = 6 \) (Equation 2)

Solve Equation 1 for \( x \):
\( x + 6 = 5 \)
\( x = 5 - 6 \)
\( x = -1 \)

Verify using Equation 2:
\( 4(-1) + 10 = 6 \)
\( -4 + 10 = 6 \)
\( 6 = 6 \), which is true.

Thus, the value of \( x \) is -1.