Problem 21 - 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 \(A^{-1} A=I\) $$ A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ -2 & 0 & 1 \\ 1 & -1 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse matrix \(A^{-1}\) is \[ A^{-1} = \left[\begin{array}{rrr} -1 & 2 & -1 \ 1 & 4 & -2 \ -2 & 2 & 1 \end{array}\right] \]

Step 1: Setup

Start by setting up [A | I], where A is the given matrix and I is the 3x3 identity matrix. \[ [ A | I ] = \left[\begin{array}{rrr|rrr} 1 & 2 & -1 & 1 & 0 & 0 \ -2 & 0 & 1 & 0 & 1 & 0 \ 1 & -1 & 0 & 0 & 0 & 1 \end{array}\right] \]

Step 2: Row Operations

Perform row operations to obtain [I | B]. For example, starting with replacing Row 2 with Row 2 + 2*Row 1. Then multiply Row 1 by 2 and add Row 3 and replace Row 3. Keep manipulating the rows until left side becomes 3x3 identity matrix. \[ [ I | B ] = \left[\begin{array}{rrr|rrr} 1 & 0 & 0 & -1 & 2 & -1 \ 0 & 1 & 0 & 1 & 4 & -2 \ 0 & 0 & 1 & -2 & 2 & 1 \end{array}\right] \]

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

The desired inverse matrix, \(A^{-1}\), is then given by B. This matrix is thus \[ A^{-1} = \left[\begin{array}{rrr} -1 & 2 & -1 \ 1 & 4 & -2 \ -2 & 2 & 1 \end{array}\right] \]

Step 4: Validation

Check that both \(A A^{-1} = I\) and \(A^{-1} A = I\). This can be done by simply performing matrix multiplication on both \(A A^{-1}\) and \(A^{-1} A\), and verify if both become identity matrices.

Key Concepts

Row Operations

Understanding row operations is crucial when finding a matrix inverse. Row operations are techniques used to manipulate the rows of a matrix. These operations include:

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

These operations can systematically simplify matrices, eventually leading to an identity matrix if the matrix is invertible.
In the context of inverting matrices, row operations are applied to the augmented matrix \([A | I]\), where \(A\) is the matrix we want to invert, and \(I\) is the identity matrix. By using row operations, the goal is to transform \([A | I]\) to \([I | B]\), where \(B\) becomes the inverse matrix.

Identity Matrix

The identity matrix is an essential concept when discussing matrix computations. In a square matrix, the identity matrix, often denoted as \(I\), is a matrix where all the elements on the main diagonal are ones, and all other elements are zeros.
For a 3x3 matrix, the identity matrix looks like this: \[I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]
The identity matrix has a unique property: any matrix multiplied by the identity matrix results in the original matrix itself. So for any matrix \(A\), the equation \(A \cdot I = A\) holds true.
When inverting a matrix, reaching an identity matrix on one side of the augmented form, \([I | B]\), indicates that \(B\) is indeed the inverse of \(A\). Furthermore, the validity of an inverse matrix \(A^{-1}\) relies on the fact that \(A \cdot A^{-1} = I\) and \(A^{-1} \cdot A = I\).

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, allowing us to combine two matrices to produce another matrix. The process involves taking rows from the first matrix and columns from the second and computing their dot products to fill the elements of the product matrix.
When checking if a computed inverse matrix \(A^{-1}\) is correct, we perform two matrix multiplications: \(A \times A^{-1}\) and \(A^{-1} \times A\).
Each of these should yield the identity matrix. This property underscores that the matrices are true inverses of each other.
For example, if you multiply the original matrix \(A\) by its supposed inverse \(B\), and the result is the identity matrix, \([A \cdot B = I]\), then \(B\) is accurately \(A^{-1}\).

3x3 Matrix Inversion

Inverting a 3x3 matrix involves a set sequence of operations, often using an augmented matrix approach, like forming \([A | I]\). This is because direct computation of an inverse can be complex.
To invert \(A\), one needs to:

• Set up the augmented matrix \([A | I]\).
• Use row operations to transform the original matrix \(A\) into the identity matrix \(I\).
• Through these operations, the identity matrix \(I\) will transform into the inverse matrix \(B\).
• Verify by checking that both \(A \cdot A^{-1} = I\) and \(A^{-1} \cdot A = I\).

This procedure reduces the complexity of inverting small matrices and relies on the fundamental properties of matrices and their identities. Thus, mastering 3x3 matrix inversion through row operations is efficient and highlights the interconnected nature of linear algebra principles.