Explanation Theorem 1: Consider A is a n × n matrix. The Determinants of the matrix A can be computed using cofactor expansion across the rows or down the columns. Show the expression for computing the Determinant of matrix A using cofactor expansion across the i t h row as follows: det A = a i 1 C i 1 + a i 2 C i 2 + ⋯ + a i n C i n (1) Here, C i j is the cofactor of matrix A . Show the expression for the cofactor of matrix A as follows C i j = ( − 1 ) i + j det A i j (2) Here, i and j are the row and column position of elements, A i j is the sub matrix obtained by deleting i t h row and j t h column of matrix A . Modify Equation (1) using Equation (3). det A = a i 1 ( − 1 ) i + 1 det A i 1 + a i 2 ( − 1 ) i + 2 det A i 2 + ⋯ + a i n ( − 1 ) i + n det A i n (3) Given information: The line passes through the point ( x 1 , y 1 ) with slope m . Refer Exercise 7. The Equation of the line passing through the point ( x 1 , y 1 ) and ( x 2 , y 2 ) is shown as follows: det [ 1 x y 1 x 1 y 1 1 x 2 y 2 ] = 0 (4) Calculation: Consider the line passes through the point ( x 1 , y 1 ) with slope m . Consider a point ( x 2 , y 2 ) also lies in the line such that ( x 2 , y 2 ) = ( x 1 + 1 , y 1 + m ) Modify Equation (4) using the points ( x 1 , y 1 ) and ( x 1 + 1 , y 1 + m ) as follows: det [ 1 x y 1 x 1 y 1 1 x 1 + 1 y 1 + m ] = 0 Consider the row 1, row 2 and row 3 of the matrix is denoted by R1 , R2 and R3. Replace R2 by R3 − R2 on left hand side of the Equation. det [ 1 x y 1 x 1 y 1 0 1 m ] R3 → R3 − R2 = 0 (5) Calculate the left hand side of Equation (5) by cofactor expansion across the first row as follows: Substitute 1 for i and 3 for n in Equation (3)

Thomas' Calculus and Linear Algebra and Its Applications Package for the Georgia Institute of Technology, 1/e

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ISBN: 9781323132098

Author: Thomas, Lay

Publisher: PEARSON C

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Chapter 3, Problem 8SE

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To find: a determinant Equation of the line passing through the points (x1,y1) and with slope m.

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Solution Summary: The author explains the determinant Equation of the line passing through the points left,y_1right, and with slope m. The Determinants of matrix A can be computed using

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To find: a determinant Equation of the line passing through the points (x1,y1) and with slope m.

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