Problem 14 - Algebra 2

Question

If matrix \(A\) has an inverse, what must be true? \(\begin{array}{llll}{\text { 1. } A A^{-1}=I} & {\text { II. } A^{-1} A=I} & {\text { III. } A^{-1} I=A^{-1}}\end{array}\) \(\begin{array}{ll}{\text { A I only }} & {\text { B II only }}\end{array}\) C) I and II only \(\quad\) D I, II, and III

Step-by-Step Solution

Answer

D) I, II, and III are true for a matrix \(A\) to have an inverse.

Step 1: Understanding matrix inverses

The definition of a matrix inverse states that for a matrix \( A \) to have an inverse \( A^{-1} \), the product of \( A \) and \( A^{-1} \) must equal the identity matrix \( I \) for both the order of multiplication \(AA^{-1}\) and \(A^{-1}A\).

Step 2: Analyzing each statement

Evaluate each provided statement. For statement I, \( AA^{-1} = I \) is true based on the definition of an inverse. For statement II, \( A^{-1}A = I \) is also true as the order of multiplication does not matter for a matrix and its inverse. For statement III, \( A^{-1}I = A^{-1} \) is simply the property of the identity matrix multiplying any matrix, not specifically concerning inverses, but it is a correct statement as well since multiplying by the identity matrix does not change the matrix.

Step 3: Selecting the correct answer

Since all three statements are true, the correct answer must include I, II, and III. Therefore, the answer is D) I, II, and III.

Key Concepts

Identity Matrix

In the world of matrices, the identity matrix plays a role akin to the number 1 in traditional multiplication. It's a special square matrix in which all the elements on the main diagonal are ones and all other elements are zeros. This unique arrangement means that when any matrix is multiplied by the identity matrix, the original matrix remains unchanged.

For instance, if we have an identity matrix, denoted by \(I\), and a square matrix \(A\), the product of the two, \(AI\) or \(IA\), will always result in matrix \(A\). The identity matrix effectively acts as a 'do nothing' operator, preserving whatever it multiplies. This property is not only fundamental to understanding matrix multiplication but also essential when discussing matrix inverses.

Matrix Multiplication

When we talk about multiplying matrices, things are a tad more complex than number multiplication. Matrix multiplication involves a row-by-column operation, meaning the elements of rows of the first matrix are multiplied with corresponding elements of columns in the second matrix and then summed up.

It's important to remember that the order in which you multiply matrices matters (it is not commutative). However, in the special case of a matrix and its inverse, the order doesn't affect the outcome; whether you multiply \(A\) by \(A^{-1}\) or \(A^{-1}\) by \(A\), the result is the identity matrix \(I\). To multiply matrices accurately, it's crucial to ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix.

Inverse Properties

The concept of an inverse in matrices is similar to taking a reciprocal in arithmetic. For a matrix \(A\) to have an inverse \(A^{-1}\), there are specific properties that must be satisfied:

• The matrix must be square (same number of rows and columns).
• When multiplied by its inverse, the result should yield the identity matrix, regardless of the multiplication order, \(AA^{-1} = A^{-1}A = I\).
• A matrix with an inverse is known as invertible or non-singular, implying that it represents a system with a unique solution.

If we express these properties in an equation, we could say that a matrix \(A\) is invertible if there exists a matrix \(B\) such that \(AB = BA = I\), where \(B\) is essentially \(A^{-1}\), the inverse of \(A\). This concept is fundamental to solving many problems in linear algebra, including system of linear equations and understanding eigenvalues and eigenspaces.