Here are the essential concepts you must grasp in order to answer the question correctly.
Determinants
A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation represented by the matrix. Determinants can be calculated using various methods, including cofactor expansion and row reduction.
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Determinants of 2×2 Matrices
Cofactor Expansion
Cofactor expansion is a method for calculating the determinant of a matrix by expressing it in terms of the determinants of smaller matrices. This involves selecting a row or column, multiplying each element by its corresponding cofactor (which is the determinant of the submatrix formed by removing the row and column of that element, adjusted by a sign), and summing these products. This technique is particularly useful for larger matrices.
Properties of Determinants
Determinants have several key properties that simplify their computation and understanding. For instance, the determinant of a product of matrices equals the product of their determinants, and swapping two rows of a matrix changes the sign of the determinant. Additionally, if a matrix has a row or column of zeros, its determinant is zero, indicating that the matrix is singular and not invertible.
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Determinants of 2×2 Matrices