Is the matrix a = [ 1 2 3 4 5 6 2 4 6 ]invertible? Check it by augmenting A with the identity matrix and trying to find the reduced row echelon form of [a|I]

To check if a matrix \( A \) is invertible, we need to ensure that the determinant is non-zero. Alternatively, we can perform Gaussian elimination to bring the matrix to its Reduced Row Echelon Form (RREF). If the matrix can be reduced to the identity matrix, then \( A \) is invertible. Otherwise, it is not.

The matrix \( A \) given is a 3x3 matrix:

\[
A = \begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
2 & 4 & 6
\end{bmatrix}
\]

Let's augment \( A \) with the identity matrix \( I \), and then attempt to reduce \( [A|I] \) to row-echelon form.

\[
[A|I] = \begin{bmatrix}
1 & 2 & 3 & 1 & 0 & 0 \\
4 & 5 & 6 & 0 & 1 & 0 \\
2 & 4 & 6 & 0 & 0 & 1
\end{bmatrix}
\]

I'll perform the row-reduction steps next.The augmented matrix \([A|I]\) reduces to:

\[
\begin{bmatrix}
1 & 0 & -1 & 0 & \frac{2}{3} & -\frac{5}{6} \\
0 & 1 & 2 & 0 & -\frac{1}{3} & \frac{2}{3} \\
0 & 0 & 0 & 1 & 0 & -\frac{1}{2}
\end{bmatrix}
\]

Notice that the third row contains all zeros in the first three columns, indicating that the original matrix \( A \) cannot be reduced to the identity matrix on the left side. Therefore, \( A \) is **not invertible**.

### Summary:
- The matrix \( A \) is not invertible because the row-reduction process fails to produce the identity matrix on the left side.
- This result also aligns with the fact that the determinant of \( A \) would be zero.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How do you compute the determinant of a matrix, and why does a zero determinant imply non-invertibility?
2. What is the geometric interpretation of a non-invertible matrix?
3. Can a matrix with linearly dependent rows or columns be invertible?
4. How does Gaussian elimination help in finding the inverse of a matrix?
5. What is the significance of the pivot columns in the RREF process?

**Tip:** If a matrix has two identical or proportional rows, it is always non-invertible.

Is the matrix a = [ 1 2 3 4 5 6 2 4 6 ] invertible? Check it by augmenting A with the identity matrix and trying to find the reduced row echelon form of [a|I]

Explore how to determine if the 3x3 matrix a = [1 2 3; 4 5 6; 2 4 6] is invertible by augmenting it with the identity matrix and performing Gaussian elimination to find its RREF. Understand key concepts like matrix inversion, determinant, and reduced row echelon form.

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