Problem 19 - 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 \(A^{-1} A=I\) $$ A=\left[\begin{array}{lll} {2} & {0} & {0} \\ {0} & {4} & {0} \\ {0} & {0} & {6} \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix A is \[A^{-1} = \left[\begin{array}{ccc}0.5 & 0 & 0 \ 0 & 0.25 & 0 \ 0 & 0 & 0.1666\end{array}\right]\]

Step 1: Form the augmented matrix

Start by creating an augmented matrix \([A | I]\). Here, I is the identity matrix.\[\left[\begin{array}{ccc|ccc}2 & 0 & 0 & 1 & 0 & 0\ 0 & 4 & 0 & 0 & 1 & 0\ 0 & 0 & 6 & 0 & 0 & 1\end{array}\right]\]

Step 2: Perform row operations

Now we are going to transform matrix A into identity matrix I using row operations. Divide the elements of the first row by 2, the second row by 4 and the third row by 6. The resulting matrix is: \[\left[\begin{array}{ccc|ccc}1 & 0 & 0 & 0.5 & 0 & 0\ 0 & 1 & 0 & 0 & 0.25 & 0\ 0 & 0 & 1 & 0 & 0 & 0.1666\end{array}\right]\]

Step 3: Extracting the inverse matrix

After performing the row operations, the augmented matrix is of the form [I | B]. With B being the result on the right side, it's the inverse of matrix A, i.e., \( A^{-1} = \left[\begin{array}{ccc}0.5 & 0 & 0 \ 0 & 0.25 & 0 \ 0 & 0 & 0.1666\end{array}\right]\)

Step 4: Verify the inverse

Lastly, we need to verify that our calculated inverse is correct. To do this, we have to multiply matrix A by its inverse, and the inverse by A, and check that both results give the identity matrix:\( AA^{-1} = \left[\begin{array}{ccc}2 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 6\end{array}\right]\cdot\left[\begin{array}{ccc}0.5 & 0 & 0 \ 0 & 0.25 & 0 \ 0 & 0 & 0.1666\end{array}\right]= \left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]= I\) And, similarly,\( A^{-1}A = \left[\begin{array}{ccc}0.5 & 0 & 0 \ 0 & 0.25 & 0 \ 0 & 0 & 0.1666\end{array}\right]\cdot\left[\begin{array}{ccc}2 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 6\end{array}\right]= \left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]= I\)Both the results are equal to the identity matrix, which confirms that the calculated matrix A^{-1} is indeed the inverse of A.

Key Concepts

Matrix Augmentation

Matrix augmentation is a useful technique when dealing with inverses and linear transformations. Here, you start with a given matrix, let's call it \( A \), and augment it with the identity matrix \( I \). The augmented matrix looks like this: \([A | I]\). This technique prepares the matrix for row operations, which will ultimately help us find the inverse. It helps by giving a clear path to transform the original matrix \( A \) into the identity matrix, enabling the flip side of the augmentation to transform into \( A^{-1} \), the inverse of our original matrix.

Row Operations

Row operations are central to the process of finding the inverse of a matrix through augmentation. These operations allow us to modify the structure of our matrix methodically. There are three key types of row operations:

• Switching two rows
• Multiplying a row by a nonzero scalar
• Adding or subtracting a multiple of one row to another row

In our step-by-step solution, we applied these operations to transform matrix \( A \) into an identity matrix. We specifically divided rows by their corresponding leading coefficients to turn them into zeroes and ones, mirroring those of the identity matrix. This systematic transformation is what ultimately reveals the inverse matrix on the augmented side.

Identity Matrix

The identity matrix \( I \) is a special kind of square matrix that plays a crucial role when dealing with matrix operations, especially inverses. It is a matrix filled with zeros except for ones along its diagonal. The identity matrix serves as the multiplicative identity in matrix algebra. For any matrix \( A \), when you multiply it by \( I \), i.e., \( A \times I \) or \( I \times A \), the result is \( A \) itself. Therefore, when changing \( A \) into \( I \) using row operations on \([A | I]\), the identity matrix helps guide us to correctly transform the identity side into the inverse of \( A \), thus achieving the configuration \([I | B]\), where \( B \) is \( A^{-1} \).

Verify Matrix Inversion

Verifying the matrix inversion is a crucial step to ensure the calculated inverse is correct. This involves multiplying the original matrix \( A \) by the purported inverse \( A^{-1} \) and checking if the product equals the identity matrix. Furthermore, you should also multiply \( A^{-1} \) by \( A \) to confirm that it similarly yields the identity matrix:

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

In our exercise, both multiplicative results equaled the identity matrix \( I \), which confirms that the derived \( A^{-1} \) is indeed the inverse. This verification step is essential as it ensures no miscalculations occurred during the process of row operations, affirming the integrity of the inverse calculation.