To find the inverse of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 7 \end{bmatrix}$, we can follow these steps:

1. Determinant of Matrix $A$

The determinant of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is calculated as: $\text{det}(A) = ad - bc$ For $A = \begin{bmatrix} 1 & 2 \\ 3 & 7 \end{bmatrix}$:

• $a = 1$
• $b = 2$
• $c = 3$
• $d = 7$

So, the determinant of $A$ is: $\text{det}(A) = (1)(7) - (2)(3) = 7 - 6 = 1$

2. Inverse of Matrix $A$

The inverse of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, if it exists, is given by: $A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ Since $\text{det}(A) = 1$, we have: $A^{-1} = \frac{1}{1} \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}$

3. Relationship Between Matrix, Determinant, and Inverse

The inverse of a matrix exists only if the determinant is non-zero. In this case, since $\text{det}(A) = 1$, which is non-zero, the inverse exists. The inverse matrix $A^{-1}$ can be used to solve systems of linear equations involving $A$. When multiplied by $A$, $A^{-1}$ yields the identity matrix, confirming it is indeed the inverse.

Would you like further clarification or have additional questions?

Here are related questions for further exploration:

1. How do you verify if the calculated inverse of a matrix is correct?
2. What happens if the determinant of a matrix is zero?
3. Can all matrices have an inverse? Why or why not?
4. How does the concept of determinants extend to larger matrices, such as 3x3 matrices?
5. What are some real-world applications of matrix inverses?

Tip: For larger matrices (e.g., 3x3), the process of finding an inverse involves more complex steps, such as calculating cofactors and adjugates.