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📏honors pre-calculus review

key term - Minor

Citation:

Definition

In the context of solving systems with Cramer's Rule, a minor is a determinant formed by deleting a row and a column from a larger determinant. Minors play a crucial role in the application of Cramer's Rule, which is a method for solving systems of linear equations by using determinants.

5 Must Know Facts For Your Next Test

  1. Minors are used in the application of Cramer's Rule to solve systems of linear equations.
  2. The minor of an element in a determinant is the determinant of the submatrix obtained by deleting the row and column containing that element.
  3. Minors are denoted by the subscript of the element they are associated with, such as $M_{ij}$ for the minor of the element in the $i$th row and $j$th column.
  4. Cofactors are related to minors, as they are the minors multiplied by $(-1)^{i+j}$, where $i$ and $j$ are the row and column indices of the element.
  5. The determinant of a matrix can be calculated by expanding it along any row or column, using the formula that involves the elements and their corresponding minors or cofactors.

Review Questions

  • Explain the role of minors in the application of Cramer's Rule to solve systems of linear equations.
    • Minors play a crucial role in Cramer's Rule, which is a method for solving systems of linear equations. Cramer's Rule expresses the solution in terms of the determinants of the coefficient matrix and matrices formed by replacing the columns of the coefficient matrix with the constants on the right-hand side of the equations. The minor of an element in the coefficient matrix is the determinant of the submatrix obtained by deleting the row and column containing that element. These minors are then used in the formulas for Cramer's Rule to calculate the values of the variables in the system of equations.
  • Describe the relationship between minors and cofactors, and explain how they are used in the calculation of determinants.
    • Minors and cofactors are closely related concepts. The cofactor of an element in a determinant is the minor of that element multiplied by $(-1)^{i+j}$, where $i$ and $j$ are the row and column indices of the element. Determinants can be calculated by expanding them along any row or column, using the formula that involves the elements and their corresponding minors or cofactors. This expansion method allows for the efficient computation of determinants, which are essential in the application of Cramer's Rule to solve systems of linear equations.
  • Analyze how the properties of minors contribute to the understanding and application of Cramer's Rule in solving systems of linear equations.
    • The properties of minors are fundamental to the understanding and application of Cramer's Rule. Minors are used to construct the matrices required for Cramer's Rule, such as the coefficient matrix and the matrices formed by replacing the columns of the coefficient matrix with the constants on the right-hand side of the equations. The determinants of these matrices, which are calculated using the minors, are then used in the formulas of Cramer's Rule to determine the values of the variables in the system of linear equations. Additionally, the relationship between minors and cofactors, and the expansion of determinants along rows or columns using minors or cofactors, provide the mathematical foundation for the efficient computation of the determinants required in Cramer's Rule.