Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix \(A\) is \(\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}\).

Step 1: Understanding the Matrix Relationship

The problem provides the inverse of matrix \(A\), denoted as \(A^{-1}\). Our goal is to find the original matrix \(A\) such that when it is multiplied with \(A^{-1}\), it results in the identity matrix \(I\). This can be expressed as: \(A \cdot A^{-1} = I\).

Step 2: Understand the Properties of Transpose in the Matrix

The given inverse matrix \(A^{-1}\) is a permutation matrix due to its single 1 per row and column. By observing the structure, notice that \(A^{-1}\) is actually the transpose of itself, i.e., \((A^{-1})^T = A\). This means the roles of row and column are swapped for the identity property.

Step 3: Transpose the Given Matrix

To find matrix \(A\), transpose \(A^{-1}\). The transpose operation involves flipping the matrix over its diagonal, switching the row and column indices. For our matrix: \[A^{-1} = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix}\] After transposition, matrix \(A\) becomes: \[A = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix}\].

Step 4: Verify the Solution

To ensure correctness, multiply \(A\) by \(A^{-1}\) and verify if it results in the identity matrix \(I\). Calculate \(A \cdot A^{-1}\) as follows: \[A \cdot A^{-1} = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\] This confirms that \(A\) is correctly identified, as it results in the 3x3 identity matrix.

Key Concepts

Matrix Multiplication

When dealing with matrices, one of the most essential operations is "Matrix Multiplication". Every time we multiply two matrices together, we create a new matrix that's a blend of the original two. This operation is not as straightforward as multiplying single numbers but involves a precise process.

Here's how it works:

• The number of columns in the first matrix must equal the number of rows in the second matrix. This is a crucial requirement for multiplication to occur.
• To find each element of the resulting matrix, multiply the elements of the rows of the first matrix by the elements of the columns of the second matrix and add up the products. For example, the first element in the product matrix is the sum of the products of the first row in the first matrix and the first column in the second matrix.

The product of a matrix with its inverse, like in our exercise, is particularly interesting because it leads to what's called the "identity matrix". This relationship highlights the inherently reversible nature of invertible matrices.

Matrix Transposition

Matrix Transposition is an operation that flips a matrix over its diagonal. This means that the rows become columns and columns become rows. It’s a simple yet powerful concept in linear algebra, critical for transformations and solving systems of equations.

When you transpose a matrix:

• The element at position (i, j) in the original matrix is moved to position (j, i) in the transposed matrix.
• For any square matrix (same number of rows and columns), the transposed version retains its size.

In our exercise, the matrix provided was essentially the transpose of the matrix we sought. This reveals a unique property of certain matrices, known as symmetric matrices, where the matrix is equal to its transpose. Recognizing such properties can simplify the journey of working with linear equations and systems.

Identity Matrix

The "Identity Matrix" is a crucial concept when studying matrices. Think of it as the matrix equivalent of the number "1" in arithmetic operations. Multiplying any number by one leaves the number unchanged, and similarly, multiplying any matrix by an identity matrix leaves it unchanged.

Here are key features of an identity matrix:

• It is a square matrix, meaning that it has the same number of rows and columns.
• The diagonal elements (from the top left to the bottom right) are all ones, while all other elements are zeros.

In the context of matrix inversion, the identity matrix plays a pivotal role because a matrix multiplied by its inverse results in the identity matrix. This property confirms that the inverse matrix correctly mirrors and cancels out the original matrix’s operations, restoring it to a neutral state.