Suppose a matrix A that reverses and shifts the values of a vector x, i.e., Ax = = (X2, X1, Xn, Xn−1, ..., X3). (a) Find A. (b) Is the matrix A invertible? Justify without using a determinant.
Suppose a matrix A that reverses and shifts the values of a vector x, i.e., Ax = = (X2, X1, Xn, Xn−1, ..., X3). (a) Find A. (b) Is the matrix A invertible? Justify without using a determinant.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Do not use Gaussian elimination/row-echelon form, numerical computation software, etc. to perform matrix inversion
![**Matrix Multiplication**
The equation illustrates the multiplication of two matrices \(A\) and \(B\), resulting in the product matrix \(AB\).
\[
AB = \begin{bmatrix} Ab_1 & Ab_2 & \cdots & Ab_n \end{bmatrix}
\]
This expression shows that the product matrix \(AB\) consists of columns that are generated by multiplying matrix \(A\) with each column vector of matrix \(B\) (\(b_1, b_2, \ldots, b_n\)).
The expanded form of the matrix multiplication is given by:
\[
= \begin{bmatrix}
(b_1)_1 a_1 + (b_1)_2 a_2 + \cdots + (b_1)_n a_n &
(b_2)_1 a_1 + \cdots + (b_2)_n a_n &
\cdots &
(b_n)_1 a_1 + \cdots + (b_n)_n a_n
\end{bmatrix}
\]
### Explanation
- Each element of the columns in the resulting matrix \(AB\) is computed as a linear combination of the columns of matrix \(A\).
- The coefficients of these combinations are the corresponding elements from columns \(b_1, b_2, \ldots, b_n\) of matrix \(B\).
This representation helps in understanding how matrix multiplication is essentially a series of linear combinations of vectors from one matrix (in this case, columns of matrix \(A\)) weighted by elements from another matrix (the columns of matrix \(B\)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1da88d38-e18c-40f1-9bb8-69bad3269ec7%2F6f672b8b-8643-4bb6-8a26-bb1b425789a5%2F3ceipkt_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Multiplication**
The equation illustrates the multiplication of two matrices \(A\) and \(B\), resulting in the product matrix \(AB\).
\[
AB = \begin{bmatrix} Ab_1 & Ab_2 & \cdots & Ab_n \end{bmatrix}
\]
This expression shows that the product matrix \(AB\) consists of columns that are generated by multiplying matrix \(A\) with each column vector of matrix \(B\) (\(b_1, b_2, \ldots, b_n\)).
The expanded form of the matrix multiplication is given by:
\[
= \begin{bmatrix}
(b_1)_1 a_1 + (b_1)_2 a_2 + \cdots + (b_1)_n a_n &
(b_2)_1 a_1 + \cdots + (b_2)_n a_n &
\cdots &
(b_n)_1 a_1 + \cdots + (b_n)_n a_n
\end{bmatrix}
\]
### Explanation
- Each element of the columns in the resulting matrix \(AB\) is computed as a linear combination of the columns of matrix \(A\).
- The coefficients of these combinations are the corresponding elements from columns \(b_1, b_2, \ldots, b_n\) of matrix \(B\).
This representation helps in understanding how matrix multiplication is essentially a series of linear combinations of vectors from one matrix (in this case, columns of matrix \(A\)) weighted by elements from another matrix (the columns of matrix \(B\)).
![1. Suppose a matrix \( A \) that reverses and shifts the values of a vector \( x \), i.e.,
\[ Ax = (x_2, x_1, x_n, x_{n-1}, \ldots, x_3). \]
(a) Find \( A \).
(b) Is the matrix \( A \) invertible? Justify *without* using a determinant.
(c) Find a matrix \( B \) such that \( AB = I \). Show that your expression for \( B \) satisfies \( AB = I \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1da88d38-e18c-40f1-9bb8-69bad3269ec7%2F6f672b8b-8643-4bb6-8a26-bb1b425789a5%2Fcp4496i_processed.png&w=3840&q=75)
Transcribed Image Text:1. Suppose a matrix \( A \) that reverses and shifts the values of a vector \( x \), i.e.,
\[ Ax = (x_2, x_1, x_n, x_{n-1}, \ldots, x_3). \]
(a) Find \( A \).
(b) Is the matrix \( A \) invertible? Justify *without* using a determinant.
(c) Find a matrix \( B \) such that \( AB = I \). Show that your expression for \( B \) satisfies \( AB = I \).
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