From 36797a4643bdee88fab6ecbd6eed8b9c1dfb1137 Mon Sep 17 00:00:00 2001 From: asset-afv <44137926+asset-afv@users.noreply.github.com> Date: Mon, 15 Oct 2018 07:11:06 +0600 Subject: [PATCH] Second method for 3 by 3 Determinants (#19090) Second method for 3 by 3 matrix Determinants, based on extended matrix diagonals --- .../mathematics/3-by-3-determinants/index.md | 167 +++++++++++++++++- 1 file changed, 166 insertions(+), 1 deletion(-) diff --git a/client/src/pages/guide/english/mathematics/3-by-3-determinants/index.md b/client/src/pages/guide/english/mathematics/3-by-3-determinants/index.md index 4538cf38a6..75f22f847e 100644 --- a/client/src/pages/guide/english/mathematics/3-by-3-determinants/index.md +++ b/client/src/pages/guide/english/mathematics/3-by-3-determinants/index.md @@ -2,7 +2,7 @@ title: 3 by 3 Determinants --- ## 3 by 3 Determinants - +### Method 1 Consider the following matrix, which we will call A: @@ -57,6 +57,171 @@ For example, consider the following matrix, which we will call B: det(B) = (-41) - 100 - 90 = -231 +### Method 2 +This method it similar to 2 by 2 determinants, and based on opertations with diagonals +Again, consider the following matrix, which we will call A: + +
+ + + + + + + + + + + + + + + +
abc
def
ghi
+ +Then the determinant of this matrix, denoted det(A), is given by: + +det(A) = a * e * i + b * f * g + c * d * h - c * e * g - f * h * a - i * b * d + +Note how three top-right to bottom-left diagonals are positive + + + + + + + + + + + + + + + + + +
a
e
i
+ + + + + + + + + + + + + + + + + +
b
f
g
+ + + + + + + + + + + + + + + + + +
c
d
h
+ +Top-left to bottom-right are negative + + + + + + + + + + + + + + + + + +
c
e
g
+ + + + + + + + + + + + + + + + + +
a
f
h
+ + + + + + + + + + + + + + + + + +
b
d
i
+ +Consider the same example as in method 2: matrix, which we will call B: + + + + + + + + + + + + + + + + + +
123
0-35
-1047
+ +det(B) is given by the formula above. We apply the formula below: + +det(B) = 1 * (-3) * 7 + 2 * 5 * (-10) + 3 * 0 * 4 - 3 * (-3) * (-10) - 5 * 4 * 1 - 7 * 2 * 0, which we simplify to: + +det(B) = -21 - 100 + 0 - 90 - 20 - 0 = -231, same, as in method 1 + + #### More information: * [Determinant of a Matrix](https://www.mathsisfun.com/algebra/matrix-determinant.html) on MathIsFun * [3x3 Determinant calculator](http://www.wolframalpha.com/widgets/view.jsp?id=7fcb0a2c0f0f41d9f4454ac2d8ed7ad6)