Compute the inverse of the matrix (it is not permitted to use determi-

nants) 0

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2 0 􀀀1 3

1 􀀀2 3 1

4 1 0 􀀀1

1 3 􀀀2 􀀀5

1

CCA

To find the inverse of the given matrix, we’ll use the row reduction method (also known as Gaussian elimination). Here’s the matrix provided:

\[
\mathbf{A} = \begin{bmatrix} 2 & 0 & -1 & 3 \\ 1 & -2 & 3 & 1 \\ 4 & 1 & 0 & -1 \\ 1 & 3 & -2 & -5 \end{bmatrix}
\]

### Steps to find the inverse:
1. **Form the augmented matrix** by placing the identity matrix next to \(\mathbf{A}\).
   
   \[
   \left[ \mathbf{A} | \mathbf{I} \right] = \begin{bmatrix} 2 & 0 & -1 & 3 & | & 1 & 0 & 0 & 0 \\ 1 & -2 & 3 & 1 & | & 0 & 1 & 0 & 0 \\ 4 & 1 & 0 & -1 & | & 0 & 0 & 1 & 0 \\ 1 & 3 & -2 & -5 & | & 0 & 0 & 0 & 1 \end{bmatrix}
   \]

2. **Use row operations** to make the left side of the augmented matrix into the identity matrix. Simultaneously apply these operations to the right side of the augmented matrix.

3. **Result**: When the left side becomes the identity matrix, the right side will transform into \(\mathbf{A}^{-1}\), the inverse of the matrix.

Let me calculate this for you.The matrix appears to be **singular** (non-invertible), which is why the calculations produce extremely large and undefined values (on the order of \(10^{15}\)). This suggests that the matrix does not have an inverse. A matrix is singular if its determinant is zero, which in practical terms means it cannot be row-reduced to the identity matrix without violating the invertibility conditions.

Would you like to explore why this matrix is singular or analyze any properties of singular matrices?

Here are 5 related questions:
1. How do we determine if a matrix is singular without using determinants?
2. What implications does singularity have on the solutions of linear equations?
3. Are there cases where a pseudo-inverse can be useful for singular matrices?
4. How do eigenvalues relate to the invertibility of a matrix?
5. What role does rank play in determining if a matrix is invertible?

**Tip:** When a matrix is singular, it often indicates dependencies among its rows or columns, meaning some rows can be expressed as linear combinations of others.

Compute the inverse of the matrix (it is not permitted to use determinants) 
\[ \begin{bmatrix} 2 & 0 & -1 & 3 \\ 1 & -2 & 3 & 1 \\ 4 & 1 & 0 & -1 \\ 1 & 3 & -2 & -5 \end{bmatrix} \]

Discover how to find the inverse of a matrix without using determinants by applying row reduction and Gaussian elimination. This example walks through the process step-by-step, discussing matrix singularity and properties of invertible matrices. Ideal for linear algebra students.

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