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To find the inverse of matrix \( M \), we need to set up a system of equations based on the definition of the inverse of a 2x2 matrix.

Given:
\[
M = \begin{pmatrix} 10 & -3 \\ -43 & 13 \end{pmatrix} \quad \text{and} \quad M^{-1} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

### Step 1: Matrix Multiplication
For \( M^{-1} \) to be the inverse of \( M \), the product \( M \times M^{-1} \) must equal the identity matrix \( I_2 \), where:
\[
I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

Let's calculate \( M \times M^{-1} \):
\[
M \times M^{-1} = \begin{pmatrix} 10 & -3 \\ -43 & 13 \end{pmatrix} \times \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

This multiplication gives us:
\[
\begin{pmatrix} 10a + (-3)c & 10b + (-3)d \\ -43a + 13c & -43b + 13d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

### Step 2: Set Up the System of Equations
From the matrix equation above, we obtain four equations:
1. \( 10a - 3c = 1 \)  (top-left entry)
2. \( 10b - 3d = 0 \)  (top-right entry)
3. \( -43a + 13c = 0 \)  (bottom-left entry)
4. \( -43b + 13d = 1 \)  (bottom-right entry)

### Step 3: Solve the System of Equations
Let's solve these equations:

1. \( 10a - 3c = 1 \)
2. \( -43a + 13c = 0 \)  implies \( c = \frac{43}{13}a \)

Substituting \( c = \frac{43}{13}a \) into the first equation:
\[
10a - 3\left(\frac{43}{13}a\right) = 1
\]
\[
\frac{130a - 129a}{13} = 1 \quad \Rightarrow \quad \frac{a}{13} = 1 \quad \Rightarrow \quad a = \frac{13}{130} = \frac{1}{10}
\]

Then \( c = \frac{43}{13} \times \frac{1}{10} = \frac{43}{130} \).

Now solve the other two equations:

1. \( 10b - 3d = 0 \)
2. \( -43b + 13d = 1 \) implies \( d = \frac{43}{13}b - \frac{1}{13} \).

Substituting \( d = \frac{43}{13}b - \frac{1}{13} \) into the first equation:
\[
10b - 3\left(\frac{43}{13}b - \frac{1}{13}\right) = 0
\]
\[
\frac{130b - 129b + 3}{13} = 0 \quad \Rightarrow \quad \frac{b}{13} = \frac{3}{130} \quad \Rightarrow \quad b = \frac{3}{10}
\]

Finally, \( d = \frac{43}{13} \times \frac{3}{10} - \frac{1}{13} = \frac{129}{130} - \frac{1}{13} = \frac{130}{130} = 1 \).

### Step 4: Write the Inverse Matrix
Thus, the inverse of matrix \( M \) is:
\[
M^{-1} = \begin{pmatrix} \frac{1}{10} & \frac{3}{10} \\ \frac{43}{130} & 1 \end{pmatrix}
\]

Would you like more details on any of these steps, or do you have any questions?

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Here are five related questions:
1. How would the determinant of \( M \) affect the existence of \( M^{-1} \)?
2. Can you derive the general formula for the inverse of a 2x2 matrix?
3. How would the system of equations change if \( M \) were a 3x3 matrix?
4. What is the geometric interpretation of finding the inverse of a matrix?
5. How can you verify that the calculated inverse matrix is correct?

**Tip:** Always check that the determinant of the matrix is non-zero before attempting to find its inverse; a zero determinant means the matrix is not invertible.

Set up a system of equations in a, b, c, d to find the inverse of matrix M = [[10, -3], [-43, 13]].

Learn how to find the inverse of a 2x2 matrix by setting up a system of linear equations. This tutorial includes step-by-step solutions for solving the equations and understanding key linear algebra concepts. Suitable for high school students in Grades 10-12.

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