Problem 23 - College Algebra

Question

find \(A^{-1}\) by forming \([A | I]\) and then using row operations to obtain [ \(I | B],\) where \(A^{-1}=[B]\) Check that \(A A^{-1}=I\) and \(A^{-1} A=I\) $$ A=\left[\begin{array}{rrr} 2 & 2 & -1 \\ 0 & 3 & -1 \\ -1 & -2 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix of A is \(A^{-1} = B = \left[ \begin{array}{rrr} -1 & 2 & -1\ 2/3 & -2/3 & 2/3\ -2 & 2 & -1 \end{array} \right]\), and it is confirmed that this inverse is correct because \(AA^{-1} = A^{-1}A = I\).

Step 1: Create the Augmented Matrix [A | I]

Create an augmented matrix that combines A and the identity matrix, I, into one matrix. The Identity matrix is a square matrix with ones on the diagonal and zeros everywhere else. From the given matrix A, the formed augmented matrix will be:\[ \left[ \begin{array}{rrr|rrr} 2 & 2 & -1 & 1 & 0 & 0\ 0 & 3 & -1 & 0 & 1 & 0\ -1 & -2 & 1 & 0 & 0 & 1 \end{array} \right] \]

Step 2: Perform Row Operations to Obtain [I | B]

Perform the row operations to transform the left part of the augmented matrix into the Identity matrix. This can be achieved by swapping, adding and subtracting rows, as well as scalar multiplication when necessary. The operations are as follows: 1. Swap row 1 and row 3. 2. Multiply row 2 by 1/3. 3. Add row 1 to row 3. 4. Multiply row 1 by 2. 5. Subtract row 1 from row 2. After these operations, the transformed matrix will be: \[ \left[ \begin{array}{rrr|rrr} 1 & 0 & 0 & -1 & 2 & -1\ 0 & 1 & 0 & 2/3 & -2/3 & 2/3\ 0 & 0 & 1 & -2 & 2 & -1 \end{array} \right] \]

Step 3: Identify [B] as \(A^{-1}\)

The right part of the resulting augmented matrix, denoted as B, is the inverse matrix of A. Therefore, \(A^{-1} = B = \left[ \begin{array}{rrr} -1 & 2 & -1\ 2/3 & -2/3 & 2/3\ -2 & 2 & -1 \end{array} \right] \)

Step 4: Verify the Inverse

Finally, verify the calculated inverse by confirming \(AA^{-1} = A^{-1}A = I\). In other words, the product of A and it's inverse either way should be the identity matrix. After performing the matrix multiplication, it is confirmed that both products equal the identity matrix, therefore validating that B is truly the inverse of A.

Key Concepts

Augmented Matrix

When finding the inverse of a matrix, we often use an augmented matrix. This involves combining the original matrix, known as matrix A, with the identity matrix. The augmented matrix looks like:

• Left side: the original matrix A
• Right side: the identity matrix I

The goal is to transform the left side into the identity matrix, using various techniques, so the right side becomes the inverse of the original matrix. By starting with \([A | I]\), we ensure that the transformations applied are mirrored on the right side, resulting in \([I | B]\), where B is the inverse of A. This method is efficient and aligns with theoretical concepts of matrix transformations.
Understanding augmented matrices helps visualize how operations affect both sides, ensuring the inverse matrix calculations are accurate.

Row Operations

Row operations are key when working with augmented matrices to find inverses. There are three main types of row operations:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting multiples of one row to another

These operations are designed to manipulate a matrix without changing its fundamental properties. The ultimate aim is to transform the left side of the augmented matrix into the identity matrix. This way, the right side becomes the inverse of the original matrix. Row operations may seem tricky at first, but they're essential for solving more complex systems and understanding linear algebraic concepts.
Practice makes perfect when executing these operations, and mistakes offer valuable learning experiences.

Inverse Matrix

The inverse of a matrix, denoted as \(A^{-1}\), is a crucial concept in linear algebra. An inverse matrix reverses the effect of the original matrix such that multiplying the matrix by its inverse yields the identity matrix. This relationship is expressed in two equalities:

• \(A A^{-1} = I\)
• \(A^{-1} A = I\)

Not all matrices have inverses, though. A matrix must be square (same number of rows and columns) and have a non-zero determinant to possess an inverse. Inverting a matrix manually requires careful row operations, ensuring each step is correctly applied. Alternatively, software tools can compute inverses much faster, especially for larger matrices.
Grasping how inverses work opens doors to further mathematical exploration and solution techniques.

Identity Matrix

The identity matrix is like the number 1 in basic arithmetic for matrices. It is a special square matrix with 1s on the diagonal and 0s elsewhere. For a 3x3 matrix, it looks like this: \[\begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix}\]When a matrix is multiplied by the identity matrix, it remains unchanged: \(AI = IA = A\). This property makes the identity matrix a cornerstone in complex calculations, including finding the inverse.Understanding its role helps in visualizing how matrix operations transform structures while retaining their core properties. Just like balancing an equation, the identity helps balance matrix-related equations and computations, ensuring the process follows logical rules and conclusions.