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Chapter 1: Problem 54

A square matrix \(A\) is said to be idempotent if \(A^{2}=A\) (a) Show that if \(A\) is idempotent, then so is \(I-A\) (b) Show that if \(A\) is idempotent, then \(2 A-I\) is invertible and is its own inverse.

Short Answer

Expert verified
(a) \(I-A\) is idempotent. (b) \(2A-I\) is invertible and its own inverse.

Step by step solution

01

Understanding the Problem

We are given that matrix A is idempotent, meaning that \( A^2 = A \). We need to show that \( I-A \) is also idempotent, and that \( 2A - I \) is invertible and is its own inverse.
02

Showing \( (I - A) \) is Idempotent

Start by computing \((I-A)^2\):\[(I-A)^2 = (I-A)(I-A) = I - A - A + A^2\]Since \( A^2 = A \) by the idempotence of \( A \), substitute \( A^2 \) with \( A \) in the expression:\[(I-A)^2 = I - 2A + A = I - A\]Thus, \( (I-A)^2 = I-A \), demonstrating that \( I-A \) is idempotent.
03

Showing \( 2A-I \) is Its Own Inverse

We need to show that \((2A-I)^2 = I\):\[(2A-I)^2 = (2A-I)(2A-I) = 4A^2 - 4A + I\]Substitute \( A^2 = A \) into the expression:\[4A^2 = 4A\]Thus, the expression simplifies:\[4A - 4A + I = I\]This verifies that \( (2A-I)^2 = I \), proving \( 2A-I \) is its own inverse.
04

Conclusion

Given that \((2A-I)^2 = I\), it shows that \(2A-I\) is invertible, and its inverse is \(2A-I\). Hence both parts of the question are solved.

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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 non-degenerate matrix, is a matrix that has an inverse. If a matrix \(A\) is invertible, there exists another matrix \(B\) such that when \(A\) is multiplied by \(B\), the result is the identity matrix. Mathematically, this is expressed as \(AB = BA = I\), where \(I\) is the identity matrix. The identity matrix for any square matrix \(A\) is a matrix that has 1's along its diagonal and 0's elsewhere. This property makes the identity matrix a special and simple matrix.

**Properties**
Invertible matrices have several important properties to note:
  • Uniqueness: The inverse of a matrix, if it exists, is unique.
  • Square Matrix Requirement: Only square matrices (matrices with equal number of rows and columns) can be invertible.
  • Determinant: An invertible matrix always has a non-zero determinant. If the determinant of a matrix is zero, it cannot be inverted.
  • Associativity: For matrices \(A, B,\) and \(C\), if \(A\) is invertible, then \((AB)C = A(BC)\).
Understanding these properties can help you in matrix computations and in determining whether or not a matrix is invertible.
Matrix Algebra
Matrix algebra is a specialized form of algebra that focuses on the study of matrices and the various operations that can be performed on them. It is a crucial area in mathematics with numerous applications in fields such as physics, engineering, computer science, and economics.

**Basic Operations**
In matrix algebra, various operations can be performed, including:
  • Addition and Subtraction: Two matrices can be added or subtracted if and only if they have the same dimensions.
  • Scalar Multiplication: Every entry of a matrix is multiplied by a scalar value.
  • Matrix Multiplication: Unlike number multiplication, matrix multiplication is not commutative, meaning that \(AB eq BA\) in general. Matrix multiplication involves taking the dot product of rows and columns.

**Advanced Concepts**
  • Transpose of a Matrix: This operation flips a matrix over its diagonal, switching the row and column indices.
  • Inverse of a Matrix: Covered under invertible matrices, it is critical in solving systems of linear equations.
  • Determinant: A special number that gives us important information about the matrix, such as whether it is invertible.
Matrix algebra forms the backbone of linear transformations and is essential in computations involving complex matrix equations.
Square Matrix
A square matrix is a matrix that has the same number of rows and columns, making it a regular 'square' shape. This equality in dimensions allows square matrices to have particular properties that are not shared by non-square matrices.

**Characteristics**
  • Diagonal Matrices: A special type of square matrix where non-diagonal elements are zero. These matrices are easy to work with because their determinant and eigenvalues can be found directly from the diagonal elements.
  • Identity Matrix: The simplest form of a square matrix, which has ones on the diagonal and zeros elsewhere. It behaves like the number 1 in multiplication.

**Importance in Mathematics**
Square matrices are key in matrix algebra as they can be involved in operations such as:
  • Finding the Inverse: Only square matrices can be invertible, as non-square matrices do not have an identity matrix of the same dimensions to work with.
  • Calculating Determinants: Determinants are solely defined for square matrices and play a significant role in assessing matrix properties, including singularity.
  • Eigenvalues and Eigenvectors: Square matrices often arise in the formulation of eigenproblems, which are central to many applications including stability analysis and quantum mechanics.
Understanding square matrices is vital as they act as building blocks in higher-dimensional vector spaces and linear transformations.

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Most popular questions from this chapter

A square matrix \(A\) is called skew-symmetric if \(A^{T}=-A\). Prove: (a) If \(A\) is an invertible skew-symmetric matrix, then \(A^{-1}\) is skew- symmetric. (b) If \(A\) and \(B\) are skew-symmetric matrices, then so are \(A^{T}, A+B, A-B,\) and \(k A\) for any scalar \(k\). (c) Every square matrix \(A\) can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. [Hint: Note the identity \(A=\frac{1}{2}\left(A+A^{T}\right)+\frac{1}{2}\left(A-A^{T}\right) \cdot ]\)

Let 0 denote a \(2 \times 2\) matrix, each of whose entries is zero. (a) Is there a \(2 \times 2\) matrix \(A\) such that \(A \neq 0\) and \(A A=0 ?\) Justify your answer. (b) Is there a \(2 \times 2\) matrix \(A\) such that \(A \neq 0\) and \(A A=A^{2}\) Justify your answer.

(a) Show that if \(A\) has a row of zeros and \(B\) is any matrix for which \(A B\) is defined, then \(A B\) also has a row of zeros. (b) Find a similar result involving a column of zeros.

In each part, express the matrix equation as a system of linear equations. a. $$\left[\begin{array}{rrr} 3 & -1 & 2 \\ 4 & 3 & 7 \\ -2 & 1 & 5 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\left[\begin{array}{r} 2 \\ -1 \\ 4 \end{array}\right]$$ b. $$\left[\begin{array}{rrrr} 3 & -2 & 0 & 1 \\ 5 & 0 & 2 & -2 \\ 3 & 1 & 4 & 7 \\ -2 & 5 & 1 & 6 \end{array}\right]\left[\begin{array}{l} w \\ x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$

Show that if $$ A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & c \end{array}\right] $$ is an elementary matrix, then at least one entry in the third row must be a zero.
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