Question

Find the inverse of the matrix if it exists. [ 1 2 3 ] [ 5-1 ] [ 1-1-10 ]

Answer 1

To find the inverse of a matrix, we need to use the augmented matrix and perform row operations to convert the left side into the identity matrix. In this case, the inverse of the given matrix is [1 0 0; -5 1 0; -1 0 1].

Step-by-step explanation:

To find the inverse of a matrix, we need to follow these steps:

1. Write the given matrix as an augmented matrix with the identity matrix on the right side.
2. Perform row operations to convert the left side of the augmented matrix into the identity matrix.
3. The resulting matrix on the right side will be the inverse of the given matrix, if it exists.

Let's apply these steps to the given matrix:

[ 1 2 3 ]

[ 5 -1 ]

[ 1 -1 -10 ]

Step 1: Write the augmented matrix

[ 1 2 3 | 1 0 0 ]

[ 5 -1 | 0 1 0 ]

[ 1 -1 -10 | 0 0 1 ]

Step 2: Perform row operations

We can perform the following row operations:

• R2 - 5R1 => R2
• R3 - R1 => R3

The resulting matrix after performing row operations:

[ 1 2 3 | 1 0 0 ]

[ 0 -11 -15 | -5 1 0 ]

[ 0 -3 -13 | -1 0 1 ]

Step 3: The right side of the augmented matrix is the inverse of the given matrix.

The inverse matrix is:

[ 1 0 0 ]

[ -5 1 0 ]

[ -1 0 1 ]