Question

Find the inverse of the matrix if it exists. [1 -2 3] [0 -1 4] [-2 2 1]

Answer 1

To find the inverse of a matrix, we can perform row operations on an augmented matrix to transform the given matrix into the identity matrix. The resulting matrix on the right side of the augmented matrix will be the inverse, if it exists.

Step-by-step explanation:

To find the inverse of a matrix, we can follow a few steps:

1. Write the matrix as [A|I], where A is the given matrix and I is the identity matrix of the same size.
2. Perform row operations on [A|I] to transform A into the identity matrix.
3. The resulting matrix on the right side of the augmented matrix will be the inverse of the given matrix, if it exists.

Let's apply these steps to the given matrix:

[1 -2 3] [0 -1 4] [-2 2 1]

[1 -2 3 | 1 0 0] [0 -1 4 | 0 1 0] [-2 2 1 | 0 0 1]

Row operations:

R1: R1 + 2R3

[1 -2 3 | 1 0 0] [0 -1 4 | 0 1 0] [0 2 7 | 0 0 1]

R3: R3 - R2

[1 -2 3 | 1 0 0] [0 -1 4 | 0 1 0] [0 3 3 | 0 -1 1]

R2: -R2

[1 -2 3 | 1 0 0] [0 1 -4 | 0 -1 0] [0 3 3 | 0 -1 1]

R2: R2 + 2R1

[1 -2 3 | 1 0 0] [0 1 -4 | 2 -1 0] [0 3 3 | 0 -1 1]

R3: R3 - 3R1

[1 -2 3 | 1 0 0] [0 1 -4 | 2 -1 0] [0 0 0 | -3 -1 1]

R3: -R3

[1 -2 3 | 1 0 0] [0 1 -4 | 2 -1 0] [0 0 0 | 3 1 -1]

R3: R3/3

[1 -2 3 | 1 0 0] [0 1 -4 | 2 -1 0] [0 0 0 | 1/3 1/3 -1/3]

R2: R2 + 4R3

[1 -2 3 | 1 0 0] [0 1 0 | 2 -1 4/3] [0 0 0 | 1/3 1/3 -1/3]

R1: R1 - 3R3

[1 -2 0 | 1 -1/3 1/3] [0 1 0 | 2 -1 4/3] [0 0 0 | 1/3 1/3 -1/3]

R1: R1 + 2R2

[1 0 0 | 5/3 -5/3 11/3] [0 1 0 | 2 -1 4/3] [0 0 0 | 1/3 1/3 -1/3]

The resulting matrix on the right is the inverse of the given matrix:

[5/3 -5/3 11/3] [2 -1 4/3] [1/3 1/3 -1/3]