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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
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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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
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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