Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 3 & 2 & 2 \\ 2 & 2 & 2 \\ -4 & 4 & 3 \end{array}\right] $$

Short answer

The inverse of the matrix is the matrix obtained on the right side of the identity matrix after doing all row operations.

Step-by-step solution

Write Down the Matrix and Augment with Identity Matrix

First, write down the given matrix and then augment it with an identity matrix on the right side. Use \( \) or \[ \] for mathematical formula. \[ [3, 2, 2; 2, 2, 2; -4, 4, 3] | I \]

Row Operations for Forming a Leading 1

Perform row operations to achieve a leading 1 in the top left corner. You can achieve this by interchanging row 1 and row 2. \[ [2, 2, 2; 3, 2, 2; -4, 4, 3] | I \]

Form Zeros Below the Leading 1

Perform row operations to create zeros below the leading 1 in the first column. We can get this by doing the following operations: R2 = R2 - 1.5*R1 and R3 = R3 +2*R1.\[ [2, 2, 2; 0, -1, -1; 0, 8, 7] | I \]

Repeat the Process for Remaining Columns

Continue the process for the remaining columns. Create a leading 1 in the second row second column and zeros below it. R2 = -1*R2 and for zeros R1 = R1 - 2*R2, R3 = R3 - 8*R2. \[ [2, 0, 0; 0, 1, 1; 0, 0, -1] | I \]

Make Leading 1 in Third Row and Finalize the Augmented Matrix

Complete the process for the third column with a leading 1 in third row and zeros in other places. R3 = -1 *R3 and R1 = R1 - 2*R3. \[ [2, 0, 0; 0, 1, 0; 0, 0, 1] | I \]

Finalize the Identity Matrix and Read off the Inverse Matrix

Finalize the Identity matrix on the left by having the first leading 1 in the first row. We can get this by R1 = 0.5*R1. \[ I | Lastly, the augmented inverse matrix after all operations is our inverse matrix.

Key concepts

Augmented Matrix

In solving for the inverse of a matrix, the concept of an augmented matrix is a fundamental starting point. An augmented matrix is essentially a combination of two matrices placed side-by-side; typically, the original matrix and another matrix of the same dimensions. In the context of finding an inverse, the original matrix is placed next to the identity matrix of the same order.

An identity matrix, denoted as 'I,' has ones on the diagonal and zeros everywhere else. Its function is similar to the number 1 in multiplication; when a matrix is multiplied by an identity matrix of the same dimensions, the original matrix is unchanged. In our exercise, the given 3x3 matrix is augmented with a 3x3 identity matrix to initiate the process of finding the inverse through a series of transformations:

• The original matrix is kept on the left side.
• The identity matrix of equal size is added on the right side.
• The resulting augmented matrix represents the system of linear equations that needs to be solved to find the inverses.

By manipulating the augmented matrix using row operations, we can transform the left side into the identity matrix, thus unveiling the inverse matrix on what was originally the right side.

Row Operations

Row operations are critical tools used to manipulate the rows of a matrix in systematic ways without changing the solutions to the system of equations represented by the matrix. There are three fundamental types of row operations:

• Row switching: One row can be swapped with another.
• Row multiplication: A row can be multiplied by a non-zero scalar.
• Row addition: A multiple of one row can be added to another row.

These operations are used to perform Gaussian elimination, a method for solving linear systems. In our exercise, row operations are used in sequence to simplify the augmented matrix. For instance, by interchanging rows to create a leading 1 in the top left corner or adjusting rows to create zeros below, we slowly transform the matrix on the left into the identity matrix. Each step in our solution — from swapping rows, multiplying them by scalars, and adding them to each other — demonstrates the application of these row operations to achieve the goal of uncovering the inverse matrix on the right side of the augmented matrix.

Identity Matrix

The identity matrix plays a pivotal role in linear algebra as a multiplicative identity for matrices. It is characterized by ones on the main diagonal and zeros in all other positions. In notation, an identity matrix of size n is often written as 'I_n' where 'n' represents its size.

The properties of the identity matrix are invaluable for many matrix operations, especially for finding the inverse of a matrix, where we aim to establish an identity matrix on one side of the augmented matrix through row operations. The identity matrix serves as a sort of 'target' or 'goal' in the process of finding the inverse. When the left side of our augmented matrix has been transformed to resemble an identity matrix, what appears on the right side is the inverse of the original matrix. In the exercise, the goal of all the row operations is to configure the left side of the augmented matrix into the identity matrix so that the right side becomes the inverse of our original matrix, thus solving the problem.