determinant

Examples of determinant in the following topics:

• Determinants of 2-by-2 Square Matrices

  • The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables.
  • It can be proven that any matrix has a unique inverse if its determinant is nonzero.
  • Various other theorems can be proved as well, including that the determinant of a product of matrices is always equal to the product of determinants; and, the determinant of a Hermitian matrix is always real.
  • In linear algebra, the determinant is a value associated with a square matrix.
  • The determinant $\begin{vmatrix} 4 & -2\\ 7 & 5 \end{vmatrix}$ is:

• Sex Determination

  • Mammalian sex is determined genetically by the presence of X and Y chromosomes .
  • Avian sex determination is dependent on the presence of Z and W chromosomes.
  • This is referred to as environmental sex determination or, more specifically, as temperature-dependent sex determination.
  • In birds, Z and W chromosomes determine sex, with females being the heterozygous sex.
  • Differentiate among the various ways animals determine the sex of offspring

• Rate-Determining Steps

  • The rate of a multi-step reaction is determined by the slowest elementary step, which is known as the rate-determining step.
  • In kinetics, the rate of a reaction with several steps is determined by the slowest step, which is known as the rate-determining, or rate-limiting, step.
  • We can picture the rate-determining step to be like the narrowest point in an hourglass; it is the "bottleneck" point of the reaction that determines how quickly reactants can become products.
  • If the second or a later step is rate-determining, determining the rate law is slightly more complicated.
  • The rate-determining step is like the narrowest point in the hourglass; it is the "bottleneck" point of the reaction that determines how quickly reactants can become products.

• Self-Determination and New States

• Cofactors, Minors, and Further Determinants

  • The determinant of any matrix can be found using its signed minors.
  • We will find the determinant of the following matrix A by calculating the determinants of its cofactors for the third, rightmost column and then multiplying them by the elements of that column.
  • As an example, we will calculate the determinant of the minor $M_{23}$, which is the determinant of the $2 \times 2$ matrix formed by removing the $2$nd row and $3$rd column.  
  • The determinant is then found by summing all of these:
  • Explain how to use minor and cofactor matrices to calculate determinants

• Molecularity

  • If a chemical reaction proceeds by more than one step or stage, its overall velocity or rate is limited by the slowest step, the rate-determining step.
  • When we describe the mechanism of a chemical reaction, it is important to identify the rate-determining step and to determine its "molecularity".
  • The molecularity of a reaction is defined as the number of molecules or ions that participate in the rate determining step.
  • A mechanism in which two reacting species combine in the transition state of the rate-determining step is called bimolecular.

• Characteristics of the Product

  • The unique characteristics of a product should be used as inputs in determining the product's marketing mix.
  • Characteristics of a product also help to determine the price of a product.
  • This can determine where a product may fall on the price index.
  • It is the combination of demand for a product and its price that help determine the marketing mix.
  • The characteristics of a product determine the target market and price of a product.

• Determining Atomic Structures by X-Ray Crystallography

  • X-ray crystallography is a method of determining the arrangement of atoms within molecules.
  • X-ray crystallography is a method for determining the arrangement of atoms within a crystal structure.
  • The steps to the process of determining the three dimensional structure of a molecule are outlined in this figure.
  • The two dimensional reflection pattern can be used to determine the atomic structure of the protein.
  • Describe the method of x-ray crystallography as it is used for determining the structure of molecules.

• Summary of the Hypothesis Test

  • Notice that in performing the hypothesis test, you use α and not β. β is needed to help determine the sample size of the data that is used in calculating the p-value.

• Structural Determination

  • Structural determination using isotopes is often performed using nuclear magnetic resonance spectroscopy and mass spectrometry.
  • Structural determination utilizing isotopes is often performed using two analytical techniques: nuclear magnetic resonance spectroscopy (NMR) and mass spectrometry (MS).
  • NMR and MS detect isotopic differences; detecting these differences allows information about the position atoms in a product's structure to be determined.
  • With information about the position of isotopic atoms in products, the reaction pathway can also be determined.
  • Identify the uses of isotopic labeling in structural determination and the primary techniques used to study isotopically-labeled molecules