Problem 22 - College Algebra

Question

In Exercises \(19-28,\) 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 \(A^{-1} A=I\) $$A=\left[\begin{array}{rrr}1 & -1 & 1 \\\0 & 2 & -1 \\\2 & 3 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse matrix \(A^{-1}=\left[\begin{array}{rrr}1 & 0 & 1 \ 0 & 1 & 1 \ -4 & -3 & 2\end{array}\right]\). Verification of the result shows that indeed \(A A^{-1}=I\) and \(A^{-1}A=I\).

Step 1: Set Up Augmented Matrix

Form the augmented matrix \([A | I]\) where \(I\) is the 3x3 identity matrix. Thus we obtain \(\left[\begin{array}{ccc|ccc}1 & -1 & 1 & 1 & 0 & 0 \ 0 & 2 & -1 & 0 & 1 & 0 \ 2 & 3 & 0 & 0 & 0 & 1\end{array}\right]\).

Step 2: Perform Row Operations

Perform row operations to transform \(A\) into the identity matrix \(I\). First, we swap row 1 and 3, then add the first row multiplied by -2 to the third row, and finally multiply the second row by 1/2. The modified matrix is \(\left[\begin{array}{ccc|ccc}1 & -1 & 1 & 1 & 0 & 0 \ 0 & 1 & -0.5 & 0 & 0.5 & 0 \ 0 & 5 & -2 & -2 & 0 & 1\end{array}\right]\). The next operation is to add row 2 to row 1 and row 2 multiplied by -5 to row 3. We obtain \(\left[\begin{array}{ccc|ccc}1 & 0 & 0.5 & 1 & 0.5 & 0 \ 0 & 1 & -0.5 & 0 & 0.5 & 0 \ 0 & 0 & 0.5 & -2 & -1.5 & 1\end{array}\right]\). At the last operation, we multiply row 3 by 2 and add it to row 1, we also multiply the third row by 2 and add it to the second row. The final matrix is \(\left[\begin{array}{ccc|ccc}1 & 0 & 0 & 1 & 0 & 1 \ 0 & 1 & 0 & 0 & 1 & 1 \ 0 & 0 & 1 & -4 & -3 & 2\end{array}\right]\).

Step 3: Finding \(A^{-1} \)

With the steps above, we were able to convert the matrix \(A\) into the identity matrix \(I\). This means that \(B\), the part of the augmented matrix that was originally \(I\), is now \(A^{-1}\).

Step 4: Verify the Inverse

Finally, verify the inverse by confirming that \(A A^{-1}=I\) and \(A^{-1}A=I\).

Key Concepts

Augmented Matrix

An augmented matrix is a fundamental tool in linear algebra for solving systems of equations and finding inverse matrices. Essentially, it combines the original matrix with another matrix, often the identity matrix, into a single larger matrix. This allows for systematic row operations to be performed on both matrices simultaneously.

For example, if you have a matrix \(A\), and you want to find its inverse \(A^{-1}\), you start by forming an augmented matrix \([A | I]\), where \(I\) is the identity matrix of the same order as \(A\). The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere, acting as the multiplicative identity in matrix operations.

• Combining matrices allows direct manipulation to transform \(A\) into \(I\).
• Row operations are used to manipulate this augmented structure.

By transforming \( [A | I] \) into \([I | B]\) via row operations, you end up with \(B\) being \(A^{-1}\). This integrated approach makes it efficient to calculate inverse matrices, especially for larger systems.

Row Operations

Row operations are manipulations performed on the rows of a matrix to achieve a desired form. They are crucial when dealing with augmented matrices to find an inverse. These operations include:

• Swapping two rows.
• Multiplying a row by a nonzero scalar.
• Adding or subtracting a multiple of one row from another row.

Each of these operations is intended to systematically transform the original matrix part of an augmented matrix into the identity matrix. For instance, if you are trying to find the inverse of matrix \(A\), row operations are used to turn \([A | I]\) into \([I | A^{-1}]\), moving \(A\) to \(I\) while transforming \(I\) into \(A^{-1}\).

These operations are powerful because, while they change the rows themselves, they do not affect the solutions of the system. This property ensures that the matrix transformations are valid and that the derived inverse is accurate.

Identity Matrix

The identity matrix plays a pivotal role in linear algebra, especially when finding inverse matrices. It is a special kind of square matrix where all the elements of the principal diagonal are ones, and all other elements are zeros.

• An \(n \times n\) identity matrix is denoted as \(I_n\).
• It behaves like the number 1 in scalar multiplication.

The identity matrix is essential because when it appears on one side of a matrix during matrix multiplication, it leaves the other matrix unchanged. For example, for any arbitrary matrix \(A\), the equation \(AI = IA = A\) holds true.

When you perform row operations on an augmented matrix \([A | I]\), the aim is to transform matrix \(A\) into \(I\). The side that was originally \(I\) in the augmented matrix will become \(A^{-1}\) by the end of these operations, effectively yielding the inverse matrix.

Matrix Multiplication

Matrix multiplication is a binary operation that takes a pair of matrices and produces another matrix. This operation is fundamental in many areas of mathematics, including finding the inverse of a matrix.
To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix has dimensions determined by the rows of the first matrix and the columns of the second. Each element in the product matrix is the dot product of a row in the first matrix with a column in the second.

• The process involves summing products of corresponding elements.
• Order matters: \(AB eq BA\) in general.

When you find an inverse matrix, verifying it involves matrix multiplication. After determining \(A^{-1}\), you confirm its correctness by checking that both \(AA^{-1} = I\) and \(A^{-1}A = I\) are true. These calculations demonstrate that the derived inverse matrix is indeed accurate, as multiplying the original matrix by its inverse yields the identity matrix, the neutral element in matrix multiplication.