Determinant of a Matrix - University of Kansas
Preview The Determinant of a SQUARE Matrix More Probelms Goals We will define determinant of SQUARE matrices, inductively, using the definition of Minors and cofactors
Determinants, part II Math 130 Linear Algebra
1 The determinant of an n n identity matrix I is 1 jIj= 1 It’s easy to check that with this construction, the determinant of the identity matrix is 1 2 If the matrix B is identical to the matrix A except the entries in one of the rows of B are each equal to the corresponding entries of A multiplied by the same scalar c, then jBj= cjAj
Math 2270 - Lecture 27 : Calculating Determinants
switches (for the second matrix the row switches are rows 1 and 3, then rows 1 and 2), and the fourth, fifth, and sixth we can get from the identity withonerow switch So, thedeterminants ofthefirst three are+1,andthe determinants of the last three are −1 Therefore, the determinant formula for a 3× 3 matrix is:
Determinants & Inverse Matrices
A matrix has an inverse exactly when its determinant is not equal to 0 ***** *** 2⇥2inverses Suppose that the determinant of the 2⇥2matrix ab cd does not equal 0 Then the matrix has an inverse, and it can be found using the formula ab cd 1 = 1 det ab cd d b ca Notice that in the above formula we are allowed to divide by the determi-
Determinants of upper/lower triangular matrices
the determinant of a triangular matrix is the product of the entries on the diagonal, detA = a 11a 22a 33:::a nn Determinants of block matrices: Block matrices are matrices of the form M = A B 0 D or M = A 0 C D with A and D square, say A is k k and D is l l and 0 - a (necessarily) l k matrix with only 0s
DETERMINANTS
4 2 3 Determinant of a matrix of order 3 × 3 Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants This is known as expansion of a determinant along a row (or a column) There are six ways of expanding a determinant of order
Determinants and eigenvalues
The determinant of a triangular matrix is the product of its diagonal entries A = 123 4 056 7 008 9 0 0 0 10 det(A)=1· 5 · 8 · 10 = 400 facts about determinantsAmazing det A can be found by “expanding” along any rowor any column
Determinants of 3×3 Matrices Date Period
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