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Chapter 2: Problem 11

Let \(A\) be an invertible \(n \times n\) matrix, and let \(B\) be an \(n \times p\) matrix. Show that the equation \(A X=B\) has a unique solution \(A^{-1} B .\)

Short Answer

Expert verified
The unique solution is \( X = A^{-1}B \).

Step by step solution

01

Understand the Equation

We are given the equation \( AX = B \), where \( A \) is an invertible \( n \times n \) matrix, \( B \) is an \( n \times p \) matrix, and \( X \) is the unknown \( n \times p \) matrix that needs to be determined.
02

Recognize the Properties of an Invertible Matrix

Since \( A \) is invertible, it has an inverse matrix \( A^{-1} \) such that \( A^{-1} A = I_n \), where \( I_n \) is the identity matrix of size \( n \times n \). This property is crucial for finding \( X \).
03

Multiply Both Sides by \( A^{-1} \)

To find \( X \), multiply both sides of the equation \( AX = B \) by \( A^{-1} \). This gives us \( A^{-1}(AX) = A^{-1}B \).
04

Simplify the Left Side of the Equation

The left side of the equation \( A^{-1}(AX) \) simplifies to \( (A^{-1}A)X \). Since \( A^{-1}A = I_n \), it simplifies further to \( I_nX = X \). Thus, we have \( X = A^{-1}B \).
05

Conclude the Solution is Unique

We have determined that \( X = A^{-1}B \). Since \( A \) is invertible and we used the inverse correctly, \( AX = B \) has one unique solution given by \( X = A^{-1}B \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Invertible Matrix
An invertible matrix, also known as a non-singular or full-rank matrix, is a square matrix that possesses an inverse. For a matrix to be invertible, it must be a square matrix, meaning it has the same number of rows and columns, typically denoted as an \( n \times n \) matrix. Here are some essential points about invertible matrices:
  • An invertible matrix has an inverse, denoted as \( A^{-1} \), which satisfies the condition \( A^{-1} A = I_n \), where \( I_n \) is the identity matrix.
  • Not every matrix is invertible; a matrix must have a non-zero determinant to qualify as invertible.
  • If \( A \) is invertible, then the solution to the matrix equation \( AX = B \) can be uniquely determined, as multiplying both sides of the equation by \( A^{-1} \) results in \( X = A^{-1}B \).
In summary, the invertibility of a matrix is a crucial property that guarantees the existence of its inverse, enabling the unique solution of certain matrix equations.
Matrix Inverse
The matrix inverse is a fundamental concept in linear algebra. When a matrix \( A \) is invertible, its inverse \( A^{-1} \) allows for operations that "undo" the effects of \( A \). The inverse satisfies several key properties:
  • The product of a matrix and its inverse results in the identity matrix, that is, \( A A^{-1} = A^{-1} A = I_n \).
  • Finding an inverse is akin to dividing a matrix, allowing you to reverse matrix multiplication when solving equations.
  • The inverse only exists for non-singular matrices, which means the matrix must have full rank and a non-zero determinant.
In practical terms, using the inverse of a matrix simplifies solving equations like \( AX = B \), as it allows us to isolate \( X \) by multiplying \( B \) with \( A^{-1} \).
Identity Matrix
An identity matrix is a special type of square matrix that plays a neutral role in matrix multiplication, similar to how the number 1 acts in regular arithmetic multiplication. Here are its primary features:
  • An identity matrix is denoted by \( I_n \), where \( n \) represents the size of the matrix, \( n \times n \).
  • It has ones on its main diagonal (from top left to bottom right) and zeros elsewhere.
  • The identity matrix behaves such that any matrix \( A \) multiplied by \( I_n \) results in \( A \) itself, i.e., \( AI_n = I_nA = A \).
The identity matrix is crucial in the concept of a matrix inverse. Specifically, when multiplying by an inverse, if \( A \) and \( A^{-1} \) are multiplied in either order, the result is the identity matrix, highlighting its role in confirming matrix invertibility and solutions.
Unique Solution
In the context of the matrix equation \( AX = B \), a unique solution refers to finding one and only one set of values for the unknown matrix \( X \) that satisfies the equation. The uniqueness of the solution is determined by certain properties of the matrix \( A \):
  • If \( A \) is invertible, it guarantees that the equation has a unique solution.
  • Multiplying \( B \) by \( A^{-1} \) solves for \( X \), resulting in \( X = A^{-1}B \), ensuring one unique answer.
  • This concept is essential in systems of linear equations, ensuring that the solution is reliable and consistent, without multiple or indefinite possibilities.
When \( A \) is invertible, there are no ambiguities about the solution. The presence of an inverse ensures that each operation effectively reverses previous steps to find the precise solution efficiently.

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Most popular questions from this chapter

In Exercises 13 and 14, find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace? $$ \left[\begin{array}{r}{1} \\ {-3} \\ {2} \\\ {-4}\end{array}\right],\left[\begin{array}{r}{-3} \\ {9} \\ {-6} \\\ {12}\end{array}\right],\left[\begin{array}{r}{2} \\ {-1} \\ {4} \\\ {2}\end{array}\right],\left[\begin{array}{r}{-4} \\ {5} \\ {-3} \\\ {7}\end{array}\right] $$

Let \(S\) be the triangle with vertices \((4.2,1.2,4),(6,4,2),\) \((2,2,6) .\) Find the image of \(S\) under the perspective projection with center of projection at \((0,0,10) .\)

Exercises \(22-26\) provide a glimpse of some widely used matrix factorizations, some of which are discussed later in the text. (Spectral Factorization) Suppose a \(3 \times 3\) matrix A admits a factorization as \(A=P D P^{-1},\) where \(P\) is some invertible \(3 \times 3\) matrix and \(D\) is the diagonal matrix $$ D=\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1 / 2} & {0} \\ {0} & {0} & {1 / 3}\end{array}\right] $$ Show that this factorization is useful when computing high powers of \(A .\) Find fairly simple formulas for \(A^{2}, A^{3},\) and \(A^{k}\) (k a positive integer), using \(P\) and the entries in \(D\) .

In Exercises 31鈥�36, respond as comprehensively as possible, and justify your answer. Suppose \(F\) is a \(5 \times 5\) matrix whose column space is not equal to \(\mathbb{R}^{5} .\) What can you say about Nul \(F ?\)

In Exercises 31鈥�36, respond as comprehensively as possible, and justify your answer. If \(P\) is a \(5 \times 5\) matrix and Nul \(P\) is the zero subspace, what can you say about solutions of equations of the form \(P \mathbf{x}=\mathbf{b}\) for \(\mathbf{b}\) in \(\mathbb{R}^{5} ?\)
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