Determinant of Triangular Matrix | Linear Algebra Proof
#triangular #linearalgebra #fullvideo
In this video, we explore the Determinant of Triangular Matrix and Linear Algebra Proof.
The determinant of an upper triangular matrix proof is shown to be the product of the diagonal entries (i.e. multiply the numbers on the main diagonal of the square matrix to find the determinant).
An upper triangular matrix (UTM) has values on the main diagonal and above (entries where the row number is less than or equal to the column number). A lower triangular matrix (LTM) has values on the main diagonal and below (entries where the row number is greater than or equal to the column number). In linear algebra, the determinant can be found using cofactor expansion of smaller determinants. Given the large number of zeros in these matrices, a lot of cancellation occurs, to produce a nice formula.
Often upper triangular matrices or lower triangular matrices are found using Gaussian Elimination on coefficients of a linear system, which is representative of a system of linear equations.
Chapters / Timestamps for Video Sections:
0:00 Intro to Upper and Lower Triangular Matrices
0:19 Algorithm / Formula determinant of UTM or LTM
0:44 Problem Statement: Prove det(A) = a_11*a_22*...*a_nn
1:16 1x1 Matrix Determinant = only entry (a_11)
1:59 2x2 Matrix Determinant = a_11*a_22 (b/c of the 0)
3:34 3x3 Matrix Determinant, cofactor expansion = a_11*a_22*a_33
6:56 Inductive Hypothesis Proof, Upper Triangular Matrix
9:38 Proof by Induction, Lower Triangular Matrix
11:05 Summary Triangular Matrices Determinants
11:50 Take action! Like! Subscribe! Share!
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Related Keywords :
Upper triangular matrix determinant,
Determinant proof,
Linear algebra concepts,
Diagonal entries of a matrix,
Matrix determinant calculation,
Triangular matrix properties,
Gaussian elimination and matrices,
Lower triangular matrix,
Cofactor expansion,
Matrix operations,
Determinant of square matrix,
Proof of determinant formula,
Hashtags :
#LinearAlgebra #MatrixDeterminant #TriangularMatrix #UpperTriangularMatrix #LowerTriangularMatrix #GaussianElimination #MatrixProof #DeterminantCalculation #MathTutorial #MatrixOperations #CofactorExpansion #SquareMatrix #MathEducation #LinearSystems #AlgebraConcepts
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