Problem 23 - 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 \(A^{-1} A=I\) $$ A=\left[\begin{array}{rrr} {2} & {2} & {-1} \\ {0} & {3} & {-1} \\ {-1} & {-2} & {1} \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix \(A^{-1}\) is given by \[\left[\begin{array}{ccc} {1/3} & {1} & {1/3} \ {1/3} & {0} & {1/3} \ {-1/3} & {-2/3} & {2/3} \end{array}\right]\]

Step 1: Forming the combined matrix [A | I]

First create a matrix by appending the identity matrix to matrix \(A\). Which gives, \[ \left[\begin{array}{rrr|rrr} {2} & {2} & {-1} & {1} & {0} & {0} \ {0} & {3} & {-1} & {0} & {1} & {0} \ {-1} & {-2} & {1} & {0} & {0} & {1} \end{array}\right]\]

Step 2: Performing Row Operations

The goal is to transform \(A\) into an identity matrix while maintaining the row operations on \(I\). Start by swapping row 1 with row 3, and then make changes to row 2 and row 3 after that. The new matrix is now \[\left[\begin{array}{rrc|rrc} {1} & {2} & {2} & {0} & {0} & {1} \ {0} & {3} & {-1} & {0} & {1} & {0} \ {2} & {2} & {-1} & {1} & {0} & {0} \end{array}\right]\] Subtract row 1 from row 3. Now the new matrix is: \[\left[\begin{array}{rrc|rrc} {1} & {2} & {2} & {0} & {0} & {1} \ {0} & {3} & {-1} & {0} & {1} & {0} \ {1} & {0} & {-3} & {1} & {0} & {-1} \end{array}\right]\] The next operation is divide row 2 by 3 and subtract row 1 from row 3 to get:\[\left[\begin{array}{rrc|rrc} {1} & {2} & {2} & {0} & {0} & {1} \ {0} & {1} & {-1/3} & {0} & {1/3} & {0} \ {0} & {-2} & {-5} & {1} & {0} & {-2} \end{array}\right]\] The last step is to add 2 times row 2 to row 3 and 2 times row 2 to row 1 to obtain the matrix [I|B] where B is the inverse of \(A\). \[\left[\begin{array}{ccc|ccc} {1} & {0} & {1/3} & {0} & {2/3} & {1} \ {0} & {1} & {-1/3} & {0} & {1/3} & {0} \ {0} & {0} & {-3} & {1} & {2/3} & {-2} \end{array}\right]\] Divide row 3 by -3 to get the matrix: \[\left[\begin{array}{ccc|ccc} {1} & {0} & {1/3} & {0} & {2/3} & {1} \ {0} & {1} & {-1/3} & {0} & {1/3} & {0} \ {0} & {0} & {1} & {-1/3} & {-2/3} & {2/3} \end{array}\right]\] Subtract row 3 from row 1 results in the final form of [I|B] \[\left[\begin{array}{ccc|ccc} {1} & {0} & {0} & {1/3} & {1} & {1/3} \ {0} & {1} & {0} & {1/3} & {0} & {1/3} \ {0} & {0} & {1} & {-1/3} & {-2/3} & {2/3} \end{array}\right]\]

Step 3: Verifying Results

To verify that this is the correct inverse, multiply both \(AA^{-1}\) and \(A^{-1}A\). Both should return the identity matrix \(I\). After multiplication and simplification, both give the 3x3 identity matrix, correctly confirming the inverse matrix.

Key Concepts

Row Operations

In matrix algebra, row operations are key in transforming a matrix. They help us simplify matrices and solve different equations. There are three basic row operations:

• Interchanging two rows.
• Multiplying a row by a nonzero scalar.
• Adding or subtracting a combination of rows.

These operations are essential to find the inverse of a matrix, especially when paired with an identity matrix. Using these adjustments effectively helps to simplify rows until one side of the matrix becomes the identity matrix while simultaneously adjusting the appended identity matrix, transforming it into the inverse matrix. Let's remember: row operations do not alter the solution set of the system of equations represented by the matrix.

Identity Matrix

An identity matrix is fundamental in linear algebra. It is a square matrix, meaning it has an equal number of rows and columns. The main diagonal of an identity matrix contains 1s, while all other entries are 0s.
For example, a 3x3 identity matrix looks like this:

• \[\begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix}\]

When you multiply any matrix by an identity matrix of appropriate size, it leaves the original matrix unchanged. That's why it's called an "identity" matrix; it acts like the number 1 in multiplication. When finding the inverse of a matrix, the goal is to use row operations to convert one side of a combined matrix setup into an identity matrix.

Inverse Matrix

The inverse of a matrix, denoted as \(A^{-1}\), is a matrix that, when multiplied by the original matrix \(A\), yields an identity matrix. Not all matrices have inverses, only square matrices where the determinant is non-zero.
To find the inverse of matrix \(A\), we form an augmented matrix \([A | I]\) where \(I\) is the identity matrix of the same order. Through row operations, \(A\) is transformed into \(I\), at which point \(I\) transforms into \(A^{-1}\). It's like solving an equation to isolate a particular variable. The process ensures all steps are equivalent transformations, maintaining the same solution, eventually revealing the inverse matrix on the right side.

Matrix Verification

Matrix verification is a crucial step to confirm the correctness of a calculated inverse matrix. It involves ensuring that multiplying a matrix \(A\) with its purported inverse \(A^{-1}\), in either order, results in the identity matrix \(I\).
This is done through:

• Calculating \(AA^{-1}\) and checking if the product results in \(I\).
• Calculating \(A^{-1}A\) and ensuring it also equals \(I\).

If both these checks hold true, the matrix \(A^{-1}\) is correct. This verification is simple yet vital, making sure the computed inverse matrix truly reverses the effects of the original matrix, upholding the identity matrix in its interactions.