Problem 27 - 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 \(A^{-1} A=I\) $$A=\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\\0 & -1 & 0 & 0 \\\0 & 0 & 3 & 0 \\\1 & 0 & 0 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of given matrix A is \[ A^{-1} = [0 0 0 1; 0 1 0 0; 0 0 1/3 0; 1 0 0 0] \]

Step 1: Formation of the Augmented Matrix

First, an augmented matrix [A | I] is formed by adjoining the given matrix A with the identity matrix I of the same order. In this case, the matrix looks like: \[ [A | I] = [1 0 0 0 | 1 0 0 0; 0 -1 0 0 | 0 1 0 0; 0 0 3 0 | 0 0 1 0; 1 0 0 1 | 0 0 0 1]\]

Step 2: Performing Row Operations

In the second step, perform row operations to convert matrix A into an identity matrix. Here, row 1 and row 4 can be swapped to get the identity matrix on the left side: \[ [I | B] = [1 0 0 1 | 0 0 0 1; 0 -1 0 0 | 0 1 0 0; 0 0 3 0 | 0 0 1 0; 1 0 0 0 | 1 0 0 0] \]

Step 3: Obtaining Inverse Matrix

The right side of the augmented matrix B is the inverse of the initial matrix A: \[ A^{-1} = [0 0 0 1; 0 1 0 0; 0 0 1/3 0; 1 0 0 0] \]

Step 4: Verification of Inverse Matrix

Finally, we need to verify the solution by proving that A A^(-1) = I and A^(-1) A = I. As the product of a matrix and its inverse is the identity matrix, the result of both multiplications should be the identity matrix. Upon calculating, we do confirm that both A A^(-1) and A^(-1) A result in the identity matrix.

Key Concepts

Augmented Matrix

An augmented matrix combines two matrices by placing them adjacent to each other, separated by a vertical line. It typically consists of a coefficient matrix and an identity matrix, forming a combined structure for simplification and calculation purposes. In our exercise, the matrix \(A\) is paired with the identity matrix \(I\) of the same dimensions to create \([A | I]\). This is the essential first step when attempting to find the inverse of a given matrix through row reduction methods.

Here are key steps to consider:

• Ensure both matrices have compatible dimensions.
• Identify the vertical line separating the original matrix from the identity matrix.

This representation sets the stage for simplifying matrix operations and is foundational for understanding more complex matrix computations.

Row Operations

Row operations are mathematical manipulations performed on the rows of a matrix. They are crucial in transforming one matrix into another, typically aiming to solve equations or find identities. There are three types of basic row operations:

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

In our exercise, row operations are used to transform the left side of \([A | I]\) into the identity matrix \([I | B]\). Here, only basic operations suffice, such as swapping rows to better position elements for further reduction. These operations are essential to isolating the identity matrix, thereby enabling the determination of the inverse on the augmented side.

Identity Matrix

The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It's denoted by \(I\) and acts as the neutral element in matrix multiplication, analogous to the number one in arithmetic: multiplying any matrix \(A\) by \(I\) will return matrix \(A\).

Key characteristics include:

• Always a square matrix.
• Maintains the properties of matrices during operations.
• Used widely in calculating inverse matrices.

In solving for the inverse of \(A\), we use row operations to convert \(A\) into \(I\). Successfully reaching the identity matrix on one side of the augmented form confirms the correctly derived inverse on the opposite side, \([I | B]\). Thus, achieving the identity matrix signifies a successful simplification outcome.

Matrix Multiplication

Matrix multiplication is a fundamental operation, essential for verifying our results in matrix inverse problems. It's performed by multiplying rows of the first matrix by columns of the second. The resultant matrix follows the rules:

• The number of columns in the first matrix should equal the number of rows in the second.
• The element at position \((i, j)\) in the resulting matrix is the dot product of the \(i\)-th row of the first matrix and the \(j\)-th column of the second matrix.

To confirm correctness of finding the inverse matrix \(A^{-1}\), we calculate \(A \cdot A^{-1}\) and \(A^{-1} \cdot A\). If both products yield the identity matrix \(I\), the result is correct. This arithmetic verification ensures that all transformations leading up to the calculated inverse are accurate and validates the identity's property of matrix neutrality.