Find the RREF of the matrix

[−41−4−3]

To find the **RREF** (Reduced Row Echelon Form) of the matrix:

\[
A = \begin{bmatrix} -4 & 1 \\ -4 & -3 \end{bmatrix}
\]

### Step-by-step Process:
1. **Write the matrix**:
   \[
   \begin{bmatrix} -4 & 1 \\ -4 & -3 \end{bmatrix}
   \]

2. **Transform the leading entry of the first row to 1**. We can divide the first row by -4:
   \[
   R_1 = \frac{1}{-4} R_1 \Rightarrow \begin{bmatrix} 1 & -\frac{1}{4} \\ -4 & -3 \end{bmatrix}
   \]

3. **Eliminate the first entry in the second row** (make it 0). We add 4 times the first row to the second row:
   \[
   R_2 = R_2 + 4R_1 \Rightarrow \begin{bmatrix} 1 & -\frac{1}{4} \\ 0 & -2 \end{bmatrix}
   \]

4. **Transform the leading entry of the second row to 1**. Divide the second row by -2:
   \[
   R_2 = \frac{1}{-2} R_2 \Rightarrow \begin{bmatrix} 1 & -\frac{1}{4} \\ 0 & 1 \end{bmatrix}
   \]

5. **Eliminate the second entry in the first row** (make it 0). Add \(\frac{1}{4}\) times the second row to the first row:
   \[
   R_1 = R_1 + \frac{1}{4} R_2 \Rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
   \]

The matrix is now in **Reduced Row Echelon Form** (RREF):
\[
\boxed{\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}}
\]

This is the identity matrix.

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Would you like more details or have any questions? 

Here are some related questions to expand on this topic:

1. What are the differences between Row Echelon Form (REF) and RREF?
2. How can you check if a matrix is invertible using its RREF?
3. What are the properties of matrices that result in an identity matrix after RREF?
4. Can non-square matrices be reduced to RREF?
5. How does RREF help solve systems of linear equations?

**Tip**: Always check the consistency of a system of equations by observing the last row in the RREF.

Find the RREF of the matrix [-4 1; -4 -3]

Learn how to find the Reduced Row Echelon Form (RREF) of the matrix [-4 1; -4 -3] using step-by-step matrix transformations. Understand key linear algebra concepts like row operations, RREF, and matrix identity transformation. Suitable for advanced high school or college-level students.

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