Problem 16. Introduction to the problem: Consider the m x n matrix A = R₁ R₂ R₁ A² = Rm C₁ a 11 a21 : ail : am1 C₂ C3 a12 a13 a22 a23 : ⠀ ai2 ⠀ am2 E Rm. C₁ A₂3 ⠀ am3 a1j a2j ⠀ aij : Amj E Rm. C₂ We can (and sometimes will) think of each row of the matrix as a vector in R" (similarly, we can think of each column in the matrix as a vector in Rm). For instance, the ith-row Ri can be thought of as the vector (a1, ai, aij,..., ain) in Rn for each 1 ≤ i ≤m (similarly, the jth-column C, can be thought of as the vector (a1j, a2j,..., aij, ..., amj) in Rm for each 1 ≤ j≤n). Cn ain The problem: Suppose that n = m, i.e., A is a square matrix of size m x m. Show that R₁. C₁ R₁.C₂ R₁ Cm R₂ C₁ R₂ C₂ R₂ Cm a2n : ain : amn. . Rm Cm
Problem 16. Introduction to the problem: Consider the m x n matrix A = R₁ R₂ R₁ A² = Rm C₁ a 11 a21 : ail : am1 C₂ C3 a12 a13 a22 a23 : ⠀ ai2 ⠀ am2 E Rm. C₁ A₂3 ⠀ am3 a1j a2j ⠀ aij : Amj E Rm. C₂ We can (and sometimes will) think of each row of the matrix as a vector in R" (similarly, we can think of each column in the matrix as a vector in Rm). For instance, the ith-row Ri can be thought of as the vector (a1, ai, aij,..., ain) in Rn for each 1 ≤ i ≤m (similarly, the jth-column C, can be thought of as the vector (a1j, a2j,..., aij, ..., amj) in Rm for each 1 ≤ j≤n). Cn ain The problem: Suppose that n = m, i.e., A is a square matrix of size m x m. Show that R₁. C₁ R₁.C₂ R₁ Cm R₂ C₁ R₂ C₂ R₂ Cm a2n : ain : amn. . Rm Cm
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question

Transcribed Image Text:Problem 16. Introduction to the problem: Consider the m x n matrix
A =
R₁
R₂
R₁
Rm
C3
a13
a23
⠀
ail
ai3
:
E
am1 am2 am3
C₁
a11
a21
:
A² =
C₂
a12
a22
:
ai2
:
Rm. C₁
...
C₂
a1j
:
Rm. C₂
azj
:
aij
:
Amj
We can (and sometimes will) think of each row of the matrix as a vector in Rn (similarly,
we can think of each column in the matrix as a vector in Rm). For instance, the ith-row Ri
can be thought of as the vector (a1, ai2,... aij, ..., ain) in Rn for each 1 ≤ i ≤m (similarly,
the jth-column C, can be thought of as the vector (a₁j, aj,..., aij,..., amj) in Rm for each
1 ≤ j<n).
The problem: Suppose that n = m, i.e., A is a square matrix of size m x m. Show that
R₁ C₁
R₁ C₂
R₁ Cm
R₂. C₁
R₂ C2
R₂. Cm
Cn
ain
a2n
:
ain
⠀
amn.
.
E
Rm. Cm
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