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

Step-by-Step Solution

Answer

The inverse of matrix A is \[A^{-1} = \left[\begin{array}{cccc}1 & 0 & 0 & 0\ 0 & -1 & 0 & 0\0 & 0 & 1/3 & 0\ -1 & 0 & 0 & 1\end{array}\right]\]

Step 1: Augment Matrix A with Identity Matrix I

Create an augmented matrix by placing matrix A and the Identity Matrix I side-by-side. \[Augmented Matrix = \left[\begin{array}{cccc|cccc}1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\ 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0\0 & 0 & 3 & 0 & 0 & 0 & 1 & 0\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1\end{array}\right]\]

Step 2: Perform Row Operations

Subtract row 1 from row 4 to obtain the identity matrix on the left side of the augmented matrix:\[\left[\begin{array}{cccc|cccc}1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\ 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0\0 & 0 & 3 & 0 & 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1\end{array}\right]\]and then multiply the second row by -1 and the third by 1/3 to get identity matrix on the left side:\[\left[\begin{array}{cccc|cccc}1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\ 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0\0 & 0 & 1 & 0 & 0 & 0 & 1/3 & 0\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1\end{array}\right]\]

Step 3: Identify the Matrix B

We now see that we have transformed our augmented matrix into [I|B], where B is our desired inverse A^-1:\[A^{-1} = B = \left[\begin{array}{cccc}1 & 0 & 0 & 0\ 0 & -1 & 0 & 0\0 & 0 & 1/3 & 0\ -1 & 0 & 0 & 1\end{array}\right]\]

Step 4: Check validity of the solution

For complete confidence of the solution, multiplication of A and A^-1, in both ways, should result in an identity matrix I:\[AA^{-1} = \left[\begin{array}{cccc}1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\0 & 0 & 1 & 0\ 0 & 0 & 0 & 1\end{array}\right] = I\]and\[A^{-1}A = \left[\begin{array}{cccc}1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\0 & 0 & 1 & 0\ 0 & 0 & 0 & 1\end{array}\right] = I\]Since both products equal the identity matrix, we can be sure that our solution is correct.

Key Concepts

Row Operations

Understanding row operations is crucial when working with matrices, especially for tasks like finding the inverse of a matrix. Row operations include swapping two rows, multiplying a row by a scalar (any number), and adding or subtracting the multiples of rows from each other. These operations can be used to simplify matrices into more manageable forms without changing the essential properties.

For example, if you need to transform a matrix to its row-echelon form or reduced row-echelon form, you'll apply a series of row operations. Specifically, when finding the inverse of a matrix, these operations help in converting the given matrix to the identity matrix, which is a significant step in the process.

Augmented Matrix

An augmented matrix combines two matrices side-by-side and is most often used in the context of solving systems of linear equations or finding the inverse of a matrix, as in our example. Typically, you write the coefficients of the variables in the system as the left part of the matrix and the constants as the right side.

In our case of finding the inverse of matrix A, we augment it with the identity matrix I. This essentially places the matrix A and I in one wider matrix, which sets the stage for applying row operations to achieve the form where the left side becomes the identity matrix, hence revealing the inverse of A on the right.

Identity Matrix

The identity matrix, usually denoted as I, is a square matrix with ones on the diagonal and zeros elsewhere. Its main property is that when it is multiplied by any compatible matrix, the result is that very same matrix. In other words, the identity matrix serves as the multiplicative identity in matrix operations, similar to the number 1 for real numbers.

The presence of the identity matrix is essential when solving for the inverse because one of our goals is to transform the original matrix to an identity matrix through row operations. Achieving this confirms we are on the right track to find the inverse, as illustrated in the given exercise.

Inverse of a Matrix

The inverse of a matrix A, denoted as A^-1, is a matrix that, when multiplied by A, results in the identity matrix. Not every matrix has an inverse—only those that are square (same number of rows and columns) and non-singular (having a non-zero determinant) do. The process to find the inverse involves creating an augmented matrix with the identity matrix, then performing row operations to convert the original matrix into the identity matrix. The result on the side of what was initially the identity matrix is then the inverse of A.

In practical terms, having the inverse of a matrix allows for solving matrix equations and can be particularly helpful in various fields such as physics, computer graphics, and more generally wherever linear transformations are involved.