Explanation Given: Given is the matrix A = [ a 1 , a 2 , a 3 , .... , a n ] be an n × n matrix, and let b = c 1 a 1 + c 2 a 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + c n a n , where c 1 , c 2 , ...... , c n are constant. Approach: P 6 property of matrix says that a matrix satisfy E = A + B then det ( E ) = det ( A ) + det ( B ) . Calculation: Given is the matrix A = [ a 1 , a 2 , a 3 , .... , a n ] be an n × n matrix, and let b = c 1 a 1 + c 2 a 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + c n a n , where c 1 , c 2 , ...... , c n are constant. So, B k can be written as B k = [ a 1 , a 2 , a k − 1 , c 1 a 1 + c 2 a 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + c n a n , a k + 1 , .... , a n ] . So, determinant of this matrix gives det ( B k ) = det ( [ a 1 , a 2 , a k − 1 , c 1 a 1 + c 2 a 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + c n a n , a k , .... , a n ] ) . Using property P 6 , det ( B k ) = det ( [ a 1 , a 2 , a k − 1 , c 1 a 1 , a k + 1 ,

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ISBN: 8220102019799

Author: ANNIN

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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…

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Chapter 3.2, Problem 60P

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To prove:

det(Bk)=ckdet(A),k=1,2,....,n holds true if Bk denotes the matrix obtained from A by replacing the kth column vector by b.

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Chapter 3.2, Problem 59P

Chapter 3.2, Problem 61P

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EBK DIFFERENTIAL EQUATIONS AND LINEAR A

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To prove: det ( B k ) = c k det ( A ) , k = 1 , 2 , .... , n holds true if B k denotes the matrix obtained from A by replacing the k th column vector by b .

Solution Summary: The author explains how P6 property of matrix says that a matrix satisfy E=A+B by replacing the kth column vector.

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To prove:

det(Bk)=ckdet(A),k=1,2,....,n holds true if Bk denotes the matrix obtained from A by replacing the kth column vector by b.

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Chapter 3 Solutions

To prove: det ( B k ) = c k det ( A ) , k = 1 , 2 , .... , n holds true if B k denotes the matrix obtained from A by replacing the k th column vector by b .

Solution Summary: The author explains how P6 property of matrix says that a matrix satisfy E=A+B by replacing the kth column vector.

Concept explainers

Videos

To prove: det(Bk)=ckdet(A),k=1,2,....,n holds true if Bk denotes the matrix obtained from A by replacing the kth column vector by b.

Want to see the full answer?

Chapter 3 Solutions

To prove:

det(Bk)=ckdet(A),k=1,2,....,n holds true if Bk denotes the matrix obtained from A by replacing the kth column vector by b.