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key term - Minor

Citation:

Definition

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

5 Must Know Facts For Your Next Test

  1. The minor of an element in a matrix is the determinant of the submatrix formed by deleting the row and column containing that element.
  2. Minors are used in the application of Cramer's Rule to solve systems of linear equations by expressing the solution in terms of the determinants of the coefficient matrix and related submatrices.
  3. The minor of an element in the coefficient matrix is denoted as $M_{ij}$, where $i$ and $j$ represent the row and column indices of the element, respectively.
  4. Minors are essential for calculating the values of the variables in a system of linear equations using Cramer's Rule, which involves dividing the determinant of a matrix formed by replacing the columns of the coefficient matrix with the constants on the right-hand side by the determinant of the coefficient matrix.
  5. The sign of a minor is determined by the row and column indices of the element, following the rule: $(-1)^{i+j}$.

Review Questions

  • Explain the relationship between minors and the determinant of a matrix in the context of Cramer's Rule.
    • In Cramer's Rule, the solution to a system of linear equations is expressed 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. The minors of the elements in the coefficient matrix are used to calculate these determinants. Specifically, the value of each variable is determined by dividing the determinant of the matrix formed by replacing the corresponding column of the coefficient matrix with the constants on the right-hand side by the determinant of the coefficient matrix. The minors play a crucial role in this process, as they are used to compute the determinants involved in Cramer's Rule.
  • Describe how the sign of a minor is determined and explain its significance in the application of Cramer's Rule.
    • The sign of a minor, $M_{ij}$, is determined by the row and column indices of the element, following the rule: $(-1)^{i+j}$. This sign is important in the application of Cramer's Rule because it affects the way the minors are combined to calculate the determinants used in the solution. The sign of the minor determines whether the contribution of the corresponding element to the determinant is positive or negative. This sign pattern is essential for correctly expressing the solution in terms of the determinants and ensuring the accuracy of the final result.
  • Analyze the role of minors in the step-by-step process of solving a system of linear equations using Cramer's Rule.
    • The use of minors is central to the application of Cramer's Rule for solving systems of linear equations. First, the determinant of the coefficient matrix is calculated, which involves the minors of its elements. Then, for each variable, a new matrix is formed by replacing the corresponding column of the coefficient matrix with the constants on the right-hand side. The determinant of this modified matrix is calculated, again using the minors of its elements. Finally, the value of each variable is determined by dividing the determinant of the modified matrix by the determinant of the coefficient matrix. Throughout this process, the minors play a crucial role in the computation of the determinants, which are the foundation of Cramer's Rule. Without a thorough understanding of minors and their properties, the successful application of Cramer's Rule would not be possible.