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Chapter 2: Problem 21

Here is an alternative argument that when \(A\) is square and \(A B=I\), it must be the case that \(B A=I\) and so \(B=A^{-1}\). a. Suppose \(A B=I\). Prove that \(A^{\top}\) is nonsingular. (Hint: Solve \(A^{\top} \mathbf{x}=\mathbf{0}\).) b. Prove there exists a matrix \(C\) so that \(A^{\top} C=I\), and hence \(C^{\top} A=I\). c. Use the result of part \(c\) of Exercise \(2.1 .11\) to prove that \(B=A^{-1}\).

Short Answer

Expert verified
We will prove that when $A$ is square and $AB=I$, it implies that $BA=I$ and $B=A^{-1}$ in three steps: a. By assuming a non-zero vector $x$ such that $A^{\top}x=0$, we reached a contradiction and concluded that $A^{\top}$ is nonsingular. b. We showed that there exists a matrix $C=A^{\top^{-1}}$ such that $A^{\top}C=I$ and $C^{\top}A=I$. c. Based on the result from part c of Exercise 2.1.11, we established that $B=A^{-1}$.

Step by step solution

01

Part (a): Prove that A岬 is nonsingular.

We have to prove that A岬 is nonsingular by solving the equation A岬x = 0. If A岬 is nonsingular, its nullspace should contain only the zero vector. Let's assume A岬 is singular, meaning there exists a non-zero vector x such that A岬x = 0. Now, we can rewrite it as A岬x = A岬 (0 * B), as AB = I (given). Now we can use the following property: A岬(BC) = (C岬A岬)B. So our equation becomes, (BA岬) x = 0. Multiplying this equation by B, we get B (BA岬)x = 0. This implies, (BBA)x = 0. Using the property (AB)A = A(BA), we have, A(BA)x = 0 or A(0) = 0, which is a contradiction. Thus, A岬 must be nonsingular.
02

Part (b): Prove the existence of matrix C.

Since A岬 is nonsingular, A岬 has an inverse denoted as A岬鈦宦 such that A岬A岬鈦宦 = I. Let C = A岬鈦宦, which gives us A岬C = I. Taking the transpose of both sides, we get (C岬A岬) = (A鈦宦贯祤A岬) = I (using the property (AB)岬 = B岬A岬). So, C岬A = I.
03

Part (c): Prove that B = A鈦宦.

According to the result of part c of Exercise 2.1.11, if AC = I and CA = I, then A鈦宦 = C. In this scenario, we have proved that A岬C = C岬A = I, so A岬鈦宦 = C = A岬鈦宦 (using A岬鈦宦 as A from the 2.1.11 result). Moreover, we already stated that C = A岬鈦宦. So, A岬鈦宦 * A岬 = I, which implies (A岬鈦宦)(A鈦解伝鹿岬鈦)岬 = I. Using the property (AB)岬 = B岬A岬, we have (A鈦宦贯祤)A岬A鈦宦 = I. Now, letting B = A鈦宦, we can rewrite it as BA = AB = I. Thus, B = A鈦宦.

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Key Concepts

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

Nonsingular Matrix
A nonsingular matrix, also known as an invertible matrix, is a square matrix that has a unique inverse. This means that when a matrix is multiplied by its inverse, the result is the identity matrix, denoted as \( I \). The identity matrix is a special kind of matrix where all elements on the main diagonal are ones, and all other elements are zeros.
  • A matrix \( A \) is nonsingular if and only if there exists a matrix \( B \) such that \( AB = BA = I \).
  • For a matrix to be nonsingular, it must also be of full rank, which means its determinant is non-zero.
In the context of the exercise, proving \( A^\top \) (the transpose of matrix \( A \)) is nonsingular involves demonstrating that its null space contains only the zero vector. This confirms that no non-zero vector \( \mathbf{x} \) satisfies \( A^\top \mathbf{x} = \mathbf{0} \), hence reinforcing the matrix鈥檚 invertibility.
Transpose of a Matrix
The transpose of a matrix, denoted \( A^\top \), is obtained by flipping the matrix over its diagonal. Essentially, the rows of the original matrix become the columns of the new matrix.
  • If a matrix \( A \) has dimensions \( m \times n \), then the transpose \( A^\top \) will have dimensions \( n \times m \).
  • This property is important for various operations in linear algebra, such as symmetric matrices where \( A = A^\top \).
In our solution, we used the concept of transpose to demonstrate properties of matrix multiplication, particularly involving nonsingular matrices. We also assumed the existence of a matrix \( C \) such that when multiplied with \( A^\top \), the identity matrix is formed, i.e., \( A^\top C = I \). This illustrates the vital role transpose plays in matrix operations.
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying rows of the first matrix by columns of the second matrix. An essential condition for multiplication is that the number of columns in the first matrix matches the number of rows in the second matrix.
  • Matrix multiplication is associative, meaning \( (AB)C = A(BC) \).
  • It is not commutative, meaning \( AB eq BA \) in general.
  • The product of any matrix and the identity matrix will yield the original matrix, \( AI = IA = A \).
In the provided solution, we use matrix multiplication to prove matrix identities and find matrix inverses. The result \( AB = I \) automatically hints that a matrix inverse exists, highlighting the significance of proper matrix order and operations.
Linear Algebra
Linear algebra is a branch of mathematics focused on vector spaces and linear mappings between these spaces. It is fundamental to various fields including computer science, engineering, and statistics.
  • Core concepts include vectors, matrices, determinants, and eigenvalues.
  • Linear algebra provides tools for solving systems of linear equations, through methods such as Gaussian elimination and matrix factorization.
  • Understanding the properties like nonsingular matrices, matrix multiplication, and transposes is essential for solving problems in linear algebra.
Through exercises like the one given, students can deepen their understanding of these concepts, particularly by proving properties and identities involving matrices. Each step enriches the comprehension of how matrix operations can be applied to real-world problems, making linear algebra a crucial tool in any mathematical toolkit.

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Most popular questions from this chapter

Let \(A_{\theta}\) be the rotation matrix defined on p. \(98,0 \leq \theta \leq \pi\). Prove that a. \(\left\|A_{\theta} \mathbf{x}\right\|=\|\mathbf{x}\|\) for all \(\mathbf{x} \in \mathbb{R}^{2}\). b. the angle between \(\mathbf{x}\) and \(A_{\theta} \mathbf{x}\) is \(\theta\). These properties characterize a rotation of the plane through angle \(\theta\).

For each of the following matrices \(A\), find a formula for \(A^{k}\) for positive integers \(k\). (If you know how to do proof by induction, please do.) a. \(A=\left[\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right]\) b. \(A=\left[\begin{array}{llll}d_{1} & & & \\ & d_{2} & & \\ & & \ddots & \\\ & & & d_{n}\end{array}\right]\) c. \(A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\)

We say an \(n \times n\) matrix \(A\) is orthogonal if \(A^{\top} A=I_{n}\). a. Prove that the column vectors \(\mathbf{a}_{1}, \ldots, \mathbf{a}_{n}\) of an orthogonal matrix \(A\) are unit vectors that are orthogonal to one another, i.e., \(\mathbf{a}_{i} \cdot \mathbf{a}_{j}= \begin{cases}1, & i=j \\ 0, & i \neq j\end{cases}\) b. Fill in the missing columns in the following matrices to make them orthogonal: $$ \left[\begin{array}{cc} \frac{\sqrt{3}}{2} & ? \\ -\frac{1}{2} & ? \end{array}\right],\left[\begin{array}{rrr} 1 & 0 & ? \\ 0 & -1 & ? \\ 0 & 0 & ? \end{array}\right], \quad\left[\begin{array}{rrr} \frac{1}{3} & ? & \frac{2}{3} \\ \frac{2}{3} & ? & -\frac{2}{3} \\ \frac{2}{3} & ? & \frac{1}{3} \end{array}\right] $$ c. Show that any \(2 \times 2\) orthogonal matrix \(A\) must be of the form $$ \left[\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] \text { or }\left[\begin{array}{rr} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array}\right] $$ for some real number \(\theta\). (Hint: Use part \(a\), rather than the original definition.) *d. Show that if \(A\) is an orthogonal \(2 \times 2\) matrix, then \(\mu_{A}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is either a rotation or the composition of a rotation and a reflection. e. Prove that the row vectors \(\mathbf{A}_{1}, \ldots, \mathbf{A}_{n}\) of an orthogonal matrix \(A\) are unit vectors that are orthogonal to one another. (Hint: Corollary 3.3.)

An \(n \times n\) matrix is called a permutation matrix if it has a single 1 in each row and column and all its remaining entries are 0 . a. Write down all the \(2 \times 2\) permutation matrices. How many are there? b. Write down all the \(3 \times 3\) permutation matrices. How many are there? c. Show that the product of two permutation matrices is again a permutation matrix. Do they commute? d. Prove that every permutation matrix is nonsingular. e. If \(A\) is an \(n \times n\) matrix and \(P\) is an \(n \times n\) permutation matrix, describe the columns of \(A P\) and the rows of \(P A\).

Find matrices \(A\) so that a. \(A \neq \mathrm{O}\), but \(A^{2}=\mathrm{O}\) b. \(A^{2} \neq \mathrm{O}\), but \(A^{3}=\mathrm{O}\) Can you make a conjecture about matrices satisfying \(A^{n-1} \neq \mathrm{O}\) but \(A^{n}=\mathrm{O}\) ?
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