Problem 26 - Algebra and Trigonometry

Question

Find \(\boldsymbol{A}^{-1}\) by forming \([\boldsymbol{A} | \boldsymbol{I}]\) and then using row operations to obtain \([I | B],\) where \(A^{-1}=[B] .\) Check that \(A A^{-1}=I\) and \(A^{-1} A=I\) $$ A=\left[\begin{array}{lll} {3} & {2} & {6} \\ {1} & {1} & {2} \\ {2} & {2} & {5} \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix \( A^{-1} \) is \( \left[ \begin{array}{ccc} 1 & 0 & -2 \ 0 & 0 & -1 \ -0.5 & 0.5 & 1 \ \end{array} \right] \). The condition \( AA^{-1} = I \) and \( A^{-1}A = I \) both hold true.

Step 1: Formulate the augmented matrix

Firstly append the identity matrix \( I \) to the given matrix \( A \) to create an augmented matrix. The resulting matrix will look like this: \( \left[ A | I \right] \) = \( \left[ \begin{array}{ccc|ccc} 3 & 2 & 6 & 1 & 0 & 0 \ 1 & 1 & 2 & 0 & 1 & 0 \ 2 & 2 & 5 & 0 & 0 & 1 \ \end{array} \right] \)

Step 2: Employ row operations

Use row operations to convert A into the identity matrix by getting the diagonal elements to be 1 and all elements above and below each pivot (the leading 1 in each row) to be 0. Applying row operations (R2 = R2 - R1 and R3 = R3 - 2*R1) gives us \( \left[ \begin{array}{ccc|ccc} 3 & 2 & 6 & 1 & 0 & 0 \ 0 & 2 & 4 & -1 & 1 & 0 \ 0 & -1 & -1 & -1 & 0 & 1 \ \end{array} \right] \) which is equivalent to our original matrix. Keep performing row operations (R2 = R2 / 2, R3 = R3 + R2 and R1 = R1 - 3/2*R2) to obtain the identity matrix on the left-hand side. The final augmented matrix might look like: \( \left[ \begin{array}{ccc|ccc} 1 & 0 & 2 & 0 & -1 & 0 \ 0 & 1 & 2 & -0.5 & 0.5 & 0 \ 0 & 0 & 1 & -0.5 & 0.5 & 1 \ \end{array} \right] \) and further performing R1 = R1 - 2*R3 will finally result in the identity matrix on the left-hand side: \( \left[ \begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & 0 & -2 \ 0 & 1 & 0 & 0 & 0 & -1 \ 0 & 0 & 1 & -0.5 & 0.5 & 1 \ \end{array} \right] \)

Step 3: Confirm the inverse

The Matrix B on the right will be \( A^{-1} \), therefore, \( A^{-1} = \left[ \begin{array}{ccc} 1 & 0 & -2 \ 0 & 0 & -1 \ -0.5 & 0.5 & 1 \ \end{array} \right] \). End this step by confirming that the multiplication of A and \( A^{-1} \), and vice versa, results in the identity matrix.

Key Concepts

Row Operations

Row operations are a fundamental technique in matrix algebra. They involve manipulating the rows of a matrix to simplify or solve linear equations. In this context, row operations help convert a matrix into its identity form.
When dealing with matrices, we perform operations such as:

• Swapping two rows: This changes the position of two rows.
• Multiplying a row by a non-zero scalar: This affects the magnitude of the entire row.
• Adding or subtracting the multiples of rows: This helps in eliminating variables during calculations.

These operations are crucial when finding the inverse of a matrix using an augmented matrix. By skilfully applying these row operations, we strive to transform the original matrix into an identity matrix, step by step.

Augmented Matrix

An augmented matrix is an extended matrix that combines two matrices to simplify operations. In this exercise, it combines the original matrix, \( A \), with the identity matrix, \( I \).
Our goal is to form a new matrix denoted as \([A | I]\). This representation makes it easier to perform row operations on \( A \) while simultaneously tracking changes on \( I \).

As the row operations progress, the left side morphs into the identity matrix. Consequently, the right side of the augmented matrix transforms into what will become the inverse matrix, \( A^{-1} \). This approach illustrates an efficient method to find matrix inverses.

Identity Matrix

The identity matrix acts like the number 1 in multiplication. For any square matrix, multiplying by the identity matrix, \( I \), gives you back the original matrix.
The identity matrix is special because it has:

• 1s on its main diagonal (from upper left to lower right).
• 0s in all other positions.

In this exercise, our aim is to turn the \( A \) side of our augmented matrix to exactly this format. Once \( A \) looks like \( I \), we know the manipulations have been applied correctly, and the right-hand side shows us the inverse matrix, \( A^{-1} \). Achieving this transformation is the ultimate goal when working with augmented matrices to find inverses.

Inverse Matrix

An inverse matrix is essential in solving systems of linear equations. For a square matrix \( A \), its inverse \( A^{-1} \) satisfies:

• \( AA^{-1} = I \)
• \( A^{-1}A = I \)

This means \( AA^{-1} \) and \( A^{-1}A \) both yield the identity matrix.
Finding \( A^{-1} \) involves using the augmented matrix and row operations, as seen in the exercise. Begin by attaching an identity matrix to \( A \), then manipulate using serial row operations until \( A \) resembles the identity matrix. At this point, the changes to \( I \) will be your inverse matrix, \( A^{-1} \).
Verifying that \( AA^{-1} \) and \( A^{-1}A \) both equal \( I \) confirms the correctness of \( A^{-1} \). This verification demonstrates successful matrix inversion.