Question

Each matrix is nonsingular. Find the inverse of each matrix. $$ \left[\begin{array}{rrr} 3 & 3 & 1 \\ 1 & 2 & 1 \\ 2 & -1 & 1 \end{array}\right] $$

Short answer

The inverse matrix is:

Step-by-step solution

- Set Up the Augmented Matrix

Start by setting up the augmented matrix combining the given matrix with the identity matrix:

- Perform Row Operations to Convert to Row Echelon Form

Use row operations to transform the left side of the augmented matrix to the identity matrix. Perform the following steps:

- Normalize the Diagonal Elements

Make the diagonal elements of the left side equal to 1 by dividing the respective rows:1. Divide Row 1 by 32. Apply further steps for Rows 2 and 3

- Eliminate Above and Below Diagonal Elements

Eliminate the elements above and below the diagonal to form a new identity matrix on the left side, completing the inversion process.

- Finalize Inverse Matrix

After performing the row operations and achieving the identity matrix on the left, the right side of the augmented matrix will be the inverse matrix. Validate the resulting matrix if desired.

Key concepts

augmented matrix

An augmented matrix is a crucial concept when finding the inverse of a matrix. Think of it as combining two matrices into a single one for easier manipulation.
Specifically, you take the given matrix and attach an identity matrix of the same size to its right side. Here's how you start:

• Write down the original matrix.
• Add a line to separate it from the identity matrix.
• Finally, write the identity matrix next to the original matrix.

For example, if you're working with the matrix:\[\begin{array}{rrr}3 & 3 & 1 \1 & 2 & 1 \2 & -1 & 1darray}\] you pair it with the identity matrix:\[\begin{array}{rrr}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1darray}\]Putting them together, the augmented matrix looks like this:\[\begin{array}{rrrrrr}3 & 3 & 1 & | & 1 & 0 & 0 \1 & 2 & 1 & | & 0 & 1 & 0 \2 & -1 & 1 & | & 0 & 0 & 1darray}\]

row operations

Row operations are actions you perform on the rows of a matrix to simplify or solve it.
They are foundational when working with matrices and involve three types:

• Swapping two rows.
• Multiplying a row by a non-zero scalar (number).
• Adding or subtracting a multiple of one row to another.

When finding the inverse of a matrix, you'll use these operations to transform the augmented matrix into a form where the left side becomes the identity matrix.
For instance, if you need to make an element 1, you might divide the row by the value of that element. Or, to eliminate an off-diagonal element, you can add or subtract a multiple of another row.
Effectively applying these row operations is what allows you to systematically convert the left side to the identity matrix while transforming the right side into the inverse matrix.

row echelon form

The objective of transforming a matrix into row echelon form is to simplify it into a shape that is easier to solve.
In row echelon form, the matrix looks like a stair step pattern where:

• All nonzero rows are above any rows of all zeros.
• The leading coefficient of each nonzero row (top leftmost non-zero number) is to the right of the leading coefficient of the row above it.
• The leading coefficient is 1, and is the only non-zero entry in its column.

Starting from the augmented matrix: \[\begin{array}{rrrrrr}3 & 3 & 1 & | & 1 & 0 & 0 \1 & 2 & 1 & | & 0 & 1 & 0 \2 & -1 & 1 & | & 0 & 0 & 1darray}\] use row operations to achieve this form. These systematic row operations will help in solving linear equations or finding inverses by gradually simplifying the matrix.

identity matrix

An identity matrix is a special kind of square matrix.
It's called an identity matrix because it acts like the number 1 in matrix multiplication. When any matrix is multiplied by an identity matrix, it remains unchanged.
Here's how you recognize it:

• All the elements on the main diagonal (from the top left to the bottom right) are 1.
• All other elements are 0.

For instance, a 3x3 identity matrix looks like this: \[\begin{array}{rrr}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1darray}\]When finding the inverse of a matrix, the goal is to manipulate the augmented matrix so that the left side transforms into the identity matrix by performing row operations.
Once this is done, the right side of the augmented matrix becomes the inverse matrix. This manipulation ensures that when the original matrix is multiplied with this new right side, the result is the identity matrix, confirming the accuracy of the inverse.