Question

Find the inverse of the matrix \[ A = \begin{pmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{pmatrix} \] by elementary transformations.

Hint:

To find the inverse of a matrix using elementary row transformations, aim to reduce the left part of the augmented matrix to the identity matrix.

Answer 1

Step 1: Augment the Matrix with the Identity Matrix

We begin by augmenting the matrix \( A \) with the identity matrix:

\[ \left[ \begin{array}{ccc|ccc} 2 & 0 & -1 & 1 & 0 & 0 \\ 5 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 3 & 0 & 0 & 1 \end{array} \right] \]

This is the augmented matrix \( [A | I] \).

Step 2: Perform Elementary Row Operations

We now perform elementary row operations to reduce the left matrix to the identity matrix. The goal is to turn \( A \) into \( I \) (the identity matrix), and apply the same operations to the identity matrix \( I \).

1. Make the pivot in the first row equal to 1

Divide row 1 by 2:

\[ R_1 \rightarrow \frac{1}{2} R_1 \] \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 5 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 3 & 0 & 0 & 1 \end{array} \right] \]

2. Eliminate the first column in rows 2 and 3

Use \( R_2 \rightarrow R_2 - 5R_1 \) and \( R_3 \rightarrow R_3 - 0R_1 \):

\[ R_2 \rightarrow R_2 - 5R_1 \] \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 1 & \frac{5}{2} & -\frac{5}{2} & 1 & 0 \\ 0 & 1 & 3 & 0 & 0 & 1 \end{array} \right] \]

3. Subtract row 2 from row 3 to eliminate the second column in row 3

\[ R_3 \rightarrow R_3 - R_2 \] \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 1 & \frac{5}{2} & -\frac{5}{2} & 1 & 0 \\ 0 & 0 & \frac{1}{2} & \frac{5}{2} & -1 & 1 \end{array} \right] \]

4. Make the pivot in row 3 equal to 1

Multiply row 3 by \( 2 \):

\[ R_3 \rightarrow 2R_3 \] \[ \left[ \begin{array}{ccc|ccc} 1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 1 & \frac{5}{2} & -\frac{5}{2} & 1 & 0 \\ 0 & 0 & 1 & 5 & -2 & 2 \end{array} \right] \]

5. Eliminate the third column in rows 1 and 2

Use \( R_1 \rightarrow R_1 + \frac{1}{2} R_3 \) and \( R_2 \rightarrow R_2 - \frac{5}{2} R_3 \):

\[ R_1 \rightarrow R_1 + \frac{1}{2} R_3, \quad R_2 \rightarrow R_2 - \frac{5}{2} R_3 \]

After these operations, we get the identity matrix on the left and the inverse matrix on the right.

\[ A^{-1} = \begin{pmatrix} \boxed{\text{Inverse Matrix}} \end{pmatrix} \]