The relation between trace and determinant of a matrix
No, there is not. Consider the matrix with parameter n [1 n n 1] The trace is 2, while the determinant is 1 − n2. You can vary n to violate any possible inequality between the trace and the determinant. Even when the matrix is diagonal: there is no relation between a sum of numbers and their product.
Are there simple methods for calculating the determinant of …
Oct 13, 2017 · Testing for a zero determinant. Look at what always happens when c=a. Disaster for invertibility. The determinant for that kind of a matrix must always be zero. When you get an equation like this for a determinant, set it equal to zero and see what happens! Those are by definition a description of all your singular matrices.
Connection between cross product and determinant
Jun 26, 2020 · 2. If →i, →j, →k are the three basic vectors of R3 then the cross product of vectors (a, b, c), (p, q, r) is the determinant of the matrix (→i →j →k a b c p q r) by definition. The coordinates of that vector are obtained by expanding this determinant along the first row. Share.
Determinant of large matrices: it must exist a faster way
May 7, 2017 · For anything larger though, it becomes absurdly complex.(use a computer, that is what they are for) There are two terms when calculating the determinant of a 2x2 matrix. There are six terms for a 3x3 matrix. For a 4x4 matrix there are 24 terms. For a 5x5 matrix, there are 120 terms. (expand by co-factors, then expand each of the 5 resulting 4x4 ...
linear algebra - Compute the characteristic equation 3x3 matrix ...
Mar 31, 2016 · The easy and quick way to compute the characteristic equation of 3x3 matrix is to use the formulae x3 − tr(A)x2 + (A11 + A22 + A33)x − det(A) = 0 For given matrix tr(A) = 4, A11(cofa11) = 3, A22(cofa22) = 1, A33(cofa33) = 1, det(A) = 2 so the char equation will be x3 − 4x2 + 5x − 2 = 0. Your formula is incorrect: A11 + A22 + A33 is the ...
What's an intuitive way to think about the determinant?
What is the determinant of the inverse of a matrix? The inverse of the determinant, of course. (Etc.) This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices.
linear algebra - Derivative of determinant of a matrix
Apr 7, 2020 · 19. The formula is d(det (m)) = det (m)Tr(m − 1dm) where dm is the matrix with dmij in the entires. The derivation is based on Cramer's rule, that m − 1 = Adj (m) det (m). It is useful in old-fashioned differential geometry involving principal bundles. I noticed Terence Tao posted a nice blog entry on it.
Expressing the determinant of a sum of two matrices?
Apr 20, 2022 · A diagonal matrix is formed with bkj. It is the Bkd below. Define also its complement Bkcd=I-Bkd. Then |X+Y|=Σk|Χ * Βkd+Y * Bkcd| It is not perfect but you can get these matrices and do transformations e.g. where matrix pencils are involved (not just characteristic polynomials in one variable, or nonlinear eigenvalues etc.)
Geometric meaning of the determinant of a matrix
Feb 10, 2016 · 41. If you think about the matrix as representing a linear transformation, then the determinant (technically the absolute value of the determinant) represents the "volume distortion" experienced by a region after being transformed. So for instance, the matrix 2I 2 I stretches a square of area 1 into a square with area 4, since the determinant is 4.
All tricks to find eigenvalues in 3x3 in a faster way?
Apr 19, 2021 · 1. For a 3 × 3 matrix, the coefficients of the characteristic polynomial are. 1, − tr(X), tr2(X) − tr(X2) 2, − det(X) which could be easier to compute. In many exercises, a solution can be found by means of the rational root theorem. In the case of three equal values on the main diagonal, you might as well have solved for λ − 1.