The problem asks to find the matrix $X$ given the equation $A \cdot X = B$, where the matrices $A$ and $B$ are:

$A = \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 8 & 2 \\ 3 & 2 \end{pmatrix}.$

Solution:

We can solve for $X$ by finding the inverse of matrix $A$ (denoted as $A^{-1}$) and then multiplying both sides of the equation by $A^{-1}$:

$A^{-1} \cdot A \cdot X = A^{-1} \cdot B \quad \Rightarrow \quad X = A^{-1} \cdot B.$

Step 1: Find $A^{-1}$

The inverse of a 2x2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the formula:

$A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},$

where $\text{det}(A) = ad - bc$ is the determinant of $A$.

For the matrix $A = \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix}$:

• $a = 2, b = 4, c = 1, d = 3$,
• $\text{det}(A) = (2)(3) - (4)(1) = 6 - 4 = 2$.

Thus, the inverse of $A$ is:

$A^{-1} = \frac{1}{2} \begin{pmatrix} 3 & -4 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} \frac{3}{2} & -2 \\ -\frac{1}{2} & 1 \end{pmatrix}.$

Step 2: Multiply $A^{-1}$ by $B$

Now, we multiply $A^{-1}$ by $B$:

$X = A^{-1} \cdot B = \begin{pmatrix} \frac{3}{2} & -2 \\ -\frac{1}{2} & 1 \end{pmatrix} \cdot \begin{pmatrix} 8 & 2 \\ 3 & 2 \end{pmatrix}.$

Let's calculate this matrix product:

$X_{11} = \left( \frac{3}{2} \times 8 \right) + \left( -2 \times 3 \right) = 12 - 6 = 6,$ $X_{12} = \left( \frac{3}{2} \times 2 \right) + \left( -2 \times 2 \right) = 3 - 4 = -1,$ $X_{21} = \left( -\frac{1}{2} \times 8 \right) + \left( 1 \times 3 \right) = -4 + 3 = -1,$ $X_{22} = \left( -\frac{1}{2} \times 2 \right) + \left( 1 \times 2 \right) = -1 + 2 = 1.$

Thus, the matrix $X$ is:

$X = \begin{pmatrix} 6 & -1 \\ -1 & 1 \end{pmatrix}.$

Final Answer:

$X = \begin{pmatrix} 6 & -1 \\ -1 & 1 \end{pmatrix}.$

Let me know if you would like further details or have questions! Here are some related questions to explore:

1. How do you compute the determinant of a 3x3 matrix?
2. Can all square matrices be inverted? Why or why not?
3. What is the geometric interpretation of the inverse of a matrix?
4. How is matrix multiplication used in solving systems of linear equations?
5. What are the properties of matrix inverses?

Tip: When calculating the inverse of a matrix, always check that the determinant is not zero. If it is, the matrix has no inverse (it's called singular).