https://file.aiquickdraw.com/mathbot/file/1727809709258ofiemnr4.jpg

The problem asks you to prove that if \( ad - bc \neq 0 \), then the matrix

\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

can be reduced to the identity matrix in row-reduced echelon form, and as a consequence, show that the system of equations

\[
\begin{aligned}
ax + by &= p \\
cx + dy &= q
\end{aligned}
\]

has a unique solution.

### Proof:

#### 1. Determinant Condition:
The determinant of the matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by

\[
\text{det} = ad - bc
\]

The condition \( ad - bc \neq 0 \) tells us that the matrix is invertible (non-singular). Therefore, we can row reduce the matrix to the identity matrix \( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \) using elementary row operations.

#### 2. Row Reduction:
We can now perform the row reduction process:

\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

**Step 1:** Make the top-left entry 1 by dividing the first row by \( a \), assuming \( a \neq 0 \):

\[
\begin{pmatrix} 1 & \frac{b}{a} \\ c & d \end{pmatrix}
\]

**Step 2:** Eliminate the lower left entry by subtracting \( c \) times the first row from the second row:

\[
\begin{pmatrix} 1 & \frac{b}{a} \\ 0 & d - \frac{bc}{a} \end{pmatrix}
\]

Since \( ad - bc \neq 0 \), we know that \( d - \frac{bc}{a} \neq 0 \), so the second pivot exists. This means we can further divide the second row by \( d - \frac{bc}{a} \) to make the pivot equal to 1.

**Step 3:** Once we complete these steps, we will have the identity matrix:

\[
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

This shows that the matrix is row-equivalent to the identity matrix.

#### 3. Solving the System of Equations:
Since the matrix can be reduced to the identity matrix, we know that the system of equations:

\[
\begin{aligned}
ax + by &= p \\
cx + dy &= q
\end{aligned}
\]

has a unique solution. Specifically, the matrix form of the system is:

\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} p \\ q \end{pmatrix}
\]

Because \( \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc \neq 0 \), the system has a unique solution, which can be found by multiplying both sides by the inverse of the matrix.

### Conclusion:
Thus, if \( ad - bc \neq 0 \), the matrix is invertible and can be row reduced to the identity matrix. As a result, the system of equations has a unique solution.

---

Do you have any questions or want further details on any step?

Here are 5 related questions to further your understanding:

1. How do you calculate the inverse of a 2x2 matrix manually?
2. What happens if \( ad - bc = 0 \)? Does the system have no solution or infinitely many solutions?
3. How does Gaussian elimination work in general for larger systems of equations?
4. Can you extend this concept to systems of equations with more than two variables?
5. How does row reduction relate to matrix rank and system solvability?

**Tip:** When solving a system of linear equations, always check the determinant of the coefficient matrix. If it's zero, the system might not have a unique solution.

Prove that if ad − bc ≠ 0, then the reduced row echelon form of the matrix is the identity matrix, and show that the system of linear equations has one unique solution.

This problem asks to prove that a 2x2 matrix is invertible if the determinant is non-zero and to show that the associated system of linear equations has one unique solution. The problem involves concepts like row reduction, matrix theory, and solving systems of equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation