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:

• 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

• Symmetry of Functions

  • In the case of symmetric functions, determining symmetry is as easy as graphing the function or evaluating the function algebraically.  
  • To determine if a relation has symmetry, graph the relation or function and see if the original curve is a reflection of itself over a point, line, or axis.  
  • Determine whether or not a given relation shows some form of symmetry

• Parts of a Hyperbola

  • The features of a hyperbola can be determined from its equation.
  • All hyperbolas share common features, and it is possible to determine the specifics of any hyperbola from the equation that defines it.
  • We will use the x-axis hyperbola to demonstrate how to determine the features of a hyperbola, so that $a$ is associated with x-coordinates and $b$ is associated with y-coordinates.
  • The major and minor axes $a$ and $b$, as the vertices and co-vertices, determine a rectangle that shares the same center as the hyperbola and has dimensions $2a \times 2b$.

• Polynomial Inequalities

  • Polynomials can be expressed as inequalities, the solutions for which can be determined from the polynomial's zeros.
  • This knowledge can then be used to determine the solutions of the inequality.  
  • We know that the lower limit of the inequality crosses the x-axis at each of these $x$ values, but now have to determine which direction (positive or negative) it takes at each crossing.
  • Recalling the initial inequality, we can now determine the solution of exactly where the polynomial is greater than zero.

• Parts of an Ellipse

  • The specific features of an ellipse can be determined from its equation.
  • We will use the horizontal case to demonstrate how to determine the properties of an ellipse from its equation, so that $a$ is associated with x-coordinates, and $b$ with y-coordinates.
  • It also shows how the sum of the distances from any point on the ellipse to the two foci is a constant, and how the eccentricity is determined by relating one of the foci to a line D called the directrix.

• The Vertical Line Test

  • The vertical line test is used to determine whether a curve on an $xy$-plane is a function
  • In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function, or not.
  • Apply the vertical line test to determine which graphs represent functions.

• Visualizing Domain and Range

  • Example 1:  Determine the domain and range of each graph pictured below:
  • Example 2: Determine the domain and range of each graph pictured below:
  • Use the graph of a function to determine its domain and range

• Cramer's Rule

  • Cramer's Rule uses determinants to solve for a solution to the equation $Ax=b$, when $A$ is a square matrix.
  • It uses a formula to calculate the solution to the system utilizing the definition of determinants.
  • It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.
  • Assume the determinant is non-zero.

• Models Involving Nonlinear Systems of Equations

  • Its position in meters (y) can be determined as a function of time in seconds (t), by the formula:
  • Its position (y) in meters can be determined as a function of time (t) in seconds, using the following formula:
  • To determine where the cars are when they are alongside one another and how much time has passed since the first began to accelerate, we can algebraically solve the system of equations using substitution: