Problem 85 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. There exists a matrix \(A\) such that \(A=A^{-1}\).

Step-by-Step Solution

Answer

Yes, it's true there exists a matrix such that \(A = A^{-1}\). Examples include the identity matrix and a diagonal matrix with \(1\) and \(-1\) on the main diagonal.

Step 1: Understand the characteristics of inverse matrices

Any pair of square matrices \(A\) and \(B\) are inverses of each other if their product is the identity matrix. That is, if \(AB=BA=I\). Here we are looking for a matrix \(A\) that's the inverse of itself. This would mean \(AA = I\).

Step 2: Consider a 2x2 identity matrix

A possible solution would be the 2x2 identity matrix itself \(I\), because multiplying it by itself still gives the identity matrix. So, the identity matrix \(I\) is a solution to \(A=A^{-1}\).

Step 3: Consider other matrices

There can be other square matrices that are their own inverse. For example, consider the 2x2 matrix \(A\) with \(1\) and \(-1\) on the diagonal and \(0\) elsewhere. Multiplying \(A\) by itself results in the identity matrix \(I\). So, this matrix \(A\) is also a solution to \(A=A^{-1}\).

Key Concepts

Identity Matrix

The identity matrix is a fundamental concept in linear algebra, often denoted as \(I\). It has unique properties that play a crucial role in operations involving matrices. The identity matrix is like the number 1 for matrix multiplication. When you multiply any square matrix by the identity matrix, you get the original matrix back.
Imagine you have a square matrix \(A\). When you multiply \(A\) by the identity matrix, the result is exactly \(A\) again. This is expressed mathematically as \(AI = IA = A\). This property makes the identity matrix an essential cornerstone when working with inverse matrices and other operations.

The identity matrix is composed of 1s on its main diagonal (from the top-left to the bottom-right) and 0s elsewhere. For example, a 2x2 identity matrix looks like this:

• \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

This pattern holds for larger matrices as well, with 1s on the diagonal and 0s on all other positions.
When thinking about the statement "\(A = A^{-1}\)", the identity matrix is an obvious candidate, because multiplying the identity matrix by itself results in the identity matrix again. This confirms that an identity matrix is indeed its own inverse.

Square Matrices

Square matrices have an equal number of rows and columns. They are significant in discussions about inverse matrices and identity matrices because they have certain properties that allow these operations to happen.
For a square matrix \(A\) to have an inverse (denoted as \(A^{-1}\)), it must be non-singular, which means its determinant is not zero. When a square matrix is multiplied by its inverse, the result is the identity matrix — indicating that the operations are reversible.

Here's why square matrices matter:

• Only square matrices can potentially have an inverse. This is because only square matrices can conform to the requirement of returning the identity matrix when multiplied by their inverse.
• The concept of a matrix being its own inverse, \(A = A^{-1}\), is exclusive to special square matrices like the identity matrix or certain matrices with specific values.

This exclusive property makes them vital in advanced matrix operations, inversions, and transformations in linear algebra.

Matrix Multiplication

Matrix multiplication is a core operation in linear algebra that enables a wide range of matrix transformations and calculations. Understanding matrix multiplication is key to comprehending concepts like identity matrices and inverses.
When you multiply two matrices, the number of columns in the first matrix must match the number of rows in the second. Each element of the resulting matrix is computed as the sum of products of corresponding elements from rows of the first matrix and columns of the second.

Benefits of matrix multiplication include:

• Transforming coordinates, which is widely used in computer graphics and data transformations.
• Combining different linear transformations into a single operation.

When considering a matrix that is its own inverse, matrix multiplication plays a central role. For example, a matrix \(A\) that satisfies \(AA = I\) showcases a situation where, through multiplication, the matrix returns the identity matrix.
To carry out successful matrix multiplication, it helps to visually map out the rows and columns or use a systematic approach to ensure each element is calculated correctly. Understanding these basic principles makes it easier to tackle complex problems utilising matrix operations.