Matrix Equation AX = B Solution

This problem requires us to solve a matrix equation to determine the value of an unknown variable, denoted by $\\alpha$. We are given three matrices, $A$, $X$, and $B$, and the relationship between them is expressed as $AX = B$. The process involves performing matrix multiplication and then solving the resulting system of linear equations.

Matrix Details Provided

The matrices provided are:

• Matrix $A = \begin{bmatrix} 3 & 7 \\ 4 & -2 \end{bmatrix}$
• Matrix $X = \begin{bmatrix} \alpha \\ -2 \end{bmatrix}$
• Matrix $B = \begin{bmatrix} 7 \\ 32 \end{bmatrix}$

The equation connecting these matrices is $AX = B$.

Matrix Multiplication AX Calculation

First, we compute the product of matrices $A$ and $X$. Matrix multiplication is performed by multiplying elements of rows from the first matrix ($A$) with corresponding elements of columns from the second matrix ($X$) and summing these products.

$AX = \begin{bmatrix} 3 & 7 \\ 4 & -2 \end{bmatrix} \begin{bmatrix} \alpha \\ -2 \end{bmatrix}$

The resulting matrix $AX$ is calculated as follows:

• The element in the first row, first column is $(3 \times \alpha) + (7 \times -2) = 3\alpha - 14$.
• The element in the second row, first column is $(4 \times \alpha) + (-2 \times -2) = 4\alpha + 4$.

Therefore, the matrix product $AX$ is:

$AX = \begin{bmatrix} 3\alpha - 14 \\ 4\alpha + 4 \end{bmatrix}$

Matrix Equality AX = B

The problem specifies that $AX = B$. We now set the calculated matrix $AX$ equal to the given matrix $B$:

$\begin{bmatrix} 3\alpha - 14 \\ 4\alpha + 4 \end{bmatrix} = \begin{bmatrix} 7 \\ 32 \end{bmatrix}$

For these two matrices to be equal, each corresponding element must be the same. This leads to a system of two linear equations:

1. First equation: $3\alpha - 14 = 7$
2. Second equation: $4\alpha + 4 = 32$

Alpha Calculation Steps

We can find the value of $\alpha$ by solving either of these equations. Let's solve the first equation:

$3\alpha - 14 = 7$

To isolate the term with $\alpha$, add 14 to both sides of the equation:

$3\alpha = 7 + 14$

$3\alpha = 21$

Now, divide both sides by 3 to find $\alpha$:

$\alpha = \frac{21}{3}$

$\alpha = 7$

To ensure consistency, let's also solve the second equation:

$4\alpha + 4 = 32$

Subtract 4 from both sides to isolate the term with $\alpha$:

$4\alpha = 32 - 4$

$4\alpha = 28$

Divide both sides by 4:

$\alpha = \frac{28}{4}$

$\alpha = 7$

Both equations consistently give $\alpha = 7$. This indicates that based on the provided matrices and the equation $AX=B$, the value of $\alpha$ is 7.

Answer Option Check

The calculation shows that $\alpha = 7$. This corresponds to Option 1. However, the provided correct answer text states that the correct option is 5 (Option 4).