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Chapter 5: Problem 5

Suppose \(A\) is a nonsingular square matrix and \(r\) is a nonzero real number. Show that \(r A\) is nonsingular and that $$ (r A)^{-1}=\frac{1}{r} A^{-1} \text {. } $$

Short Answer

Expert verified
(r A)^-1 = 1/r * A^-1.

Step by step solution

01

Show that rA is non-singular

First, since A is non-singular, it means, by definition, that its determinant is not zero. We also know that determinant of constant times matrix (rA in this case) is the constant to the power of size times the determinant of the matrix, i.e., \(det(rA) = r^n det(A)\). Here, 'n' is the size of square matrix. Since 'r' and 'A' are both non-zero, the determinant of rA is also non-zero, implying that rA is non-singular.
02

Find the inverse of rA

Consider the expression (rA)*(1/r*A^-1). Using the associativity of multiplication, this can be rewritten as [(rA)*(1/r)] * A^-1. Since r is a constant, r*1/r equals 1. Thus, the expression simplifies to: A*A^-1, which equals to the Identity matrix. Hence, 1/r * A^-1 is indeed the inverse of the matrix rA, proving that (r A)^-1 = 1/r * A^-1.

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Key Concepts

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

Nonsingular Matrices
A matrix is considered nonsingular or invertible if it has an inverse.
Nonsingular matrices are crucial in linear algebra since they allow us to perform various operations.
For a matrix to be nonsingular, its determinant must not be zero.
  • Determinant: A value calculated from the elements of a square matrix. If the determinant is zero, the matrix is singular and does not have an inverse.
  • Inverse: The matrix that, when multiplied by the original matrix, results in the identity matrix.
Understanding nonsingular matrices helps in solving systems of linear equations as they guarantee unique solutions when the matrices involved are nonsingular.
Matrix Determinant
The determinant is a critical concept when dealing with matrices, especially regarding their invertibility.
The determinant provides a scalar value representing properties of the matrix.
  • Computation: For a 2x2 matrix \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]the determinant is computed as: \[det = ad - bc\]
  • Importance: A determinant of zero indicates a singular matrix. It implies that the matrix does not have an inverse.
  • Scaling: If a matrix is multiplied by a scalar, say r, the determinant scales by \(r^n\), where \(n\) is the order of the matrix.
This scaling property is leveraged in our original exercise to demonstrate that \(rA\) is nonsingular.
Matrix Multiplication
Matrix multiplication is a binary operation that produces a new matrix from two matrices.
The rules of multiplication are vital for operations involving inversions and transformations.
  • Order: For matrix multiplication to work, the number of columns in the first matrix must equal the number of rows in the second.
  • Associative Property: This property states that \((AB)C = A(BC)\). It allows us to group matrices differently without changing the result.
  • Identity Matrix: An identity matrix is essentially the multiplicative identity for matrices, similar to 1 in numbers. Multiplying any matrix by the identity matrix results in the original matrix.
In our original exercise, the associative property and identity matrix are crucial for showing that \((rA)^{-1} = \frac{1}{r} A^{-1}\).

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Most popular questions from this chapter

a. Find four matrices \(A \in \mathbb{M}(2,2)\) with \(A^{2}=I\). b. Determine all matrices \(A \in \mathbb{M}(2,2)\) with \(A^{2}=I\).

Show that a Markov chain with two states and transition matrix \(\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) is not regular. Describe the long-term behavior of this system.

A bit of information stored in a computer can have two values, represented by 0 and 1 . Suppose that every time a bit is read, there is a very small probability \(p\) that its value will change. Use the fact (see Exercise 19 of Section \(7.1\) for a verification and Exercise 5 of Section \(8.3\) for a derivation) that $$ \left[\begin{array}{cc} 1-p & p \\ p & 1-p \end{array}\right]^{n}=\left[\begin{array}{cc} \frac{1+(1-2 p)^{n}}{2} & \frac{1-(1-2 p)^{n}}{2} \\ \frac{1-(1-2 p)^{n}}{2} & \frac{1+(1-2 p)^{n}}{2} \end{array}\right] $$ to investigate the long-term behavior of the status of the bit. Suppose a manufacturer wants to ensure that after the bit has been read 1000 times, the probability that its value will differ from the original value is less than \(.00001\). How large can \(p\) be?

Use the algorithm developed in this section to find the inverses of the following matrices (or to conclude that the inverse does not exist). a. \(\left[\begin{array}{ll}3 & 4 \\ 7 & 9\end{array}\right]\) b. \(\left[\begin{array}{rrr}1 & 2 & 1 \\ 0 & -2 & 3 \\ 4 & 1 & 2\end{array}\right]\) c. \(\left[\begin{array}{rrr}-1 & -3 & -3 \\ 1 & 2 & -1 \\ 2 & 3 & -6\end{array}\right]\) d. \(\left[\begin{array}{rrr}-1 & -3 & -3 \\ 1 & 2 & -1 \\ 2 & 3 & -3\end{array}\right]\) e. \(\left[\begin{array}{rrr}1 & -\frac{1}{2} & 0 \\ -\frac{1}{2} & 1 & -\frac{1}{2} \\ 0 & -\frac{1}{2} & 1\end{array}\right]\) f. \(\left[\begin{array}{rrr}1 & 8 & 2 \\ -2 & 3 & -1 \\ 3 & -1 & 2\end{array}\right]\)

Determine the ranks of the following matrices. a. \(\left[\begin{array}{llll}1 & 4 & 0 & 3 \\ 2 & 8 & 1 & 6 \\ 0 & 1 & 1 & 1\end{array}\right]\) b. \(\left[\begin{array}{ll}4 & 6 \\ 6 & 9 \\ 0 & 0 \\ 2 & 3\end{array}\right]\) c. \(\left[\begin{array}{llll}1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\\ 1 & 1 & 1 & 1\end{array}\right]\) d. \(\left[\begin{array}{rrrr}16 & 3 & 2 & 13 \\ 5 & 10 & 11 & 8 \\ 9 & 6 & 7 & 12 \\ 4 & 15 & 14 & 1\end{array}\right]\)
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