In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix}0.5 & 7 & 5 \\0.5 & 3 & 9 \\0.5 & 1 & 3\end{vmatrix}$

Verified step by step guidance

1. Identify the 3x3 matrix given as: \[\begin{bmatrix} 0.5 & 7 & 5 \\ 0.5 & 3 & 9 \\ 0.5 & 1 & 3 \end{bmatrix}\]
2. Choose a row or column to expand the determinant by minors. Since the first column has the same value (0.5) in all entries, expanding along the first column is a good choice.
3. Write the determinant expansion along the first column using the formula: \[\text{det}(A) = a_{11}C_{11} + a_{21}C_{21} + a_{31}C_{31}\] where \(a_{ij}\) are the elements of the matrix and \(C_{ij}\) are the cofactors.
4. Calculate each minor determinant by removing the row and column of the element you are expanding on. For example, for \(a_{11} = 0.5\), remove the first row and first column, then find the determinant of the remaining 2x2 matrix.
5. Calculate each cofactor by applying the sign pattern \((-1)^{i+j}\) to the minor determinants, then multiply each cofactor by its corresponding element and sum all three results to get the determinant.

Key Concepts

Determinant of a 3x3 Matrix

The determinant of a 3x3 matrix is a scalar value that can be computed using expansion by minors or other methods. It provides important information about the matrix, such as whether it is invertible. Calculating the determinant involves combining products of elements and their corresponding minors with alternating signs.

Expansion by Minors

Expansion by minors is a method to calculate the determinant of a matrix by selecting a row or column, then summing the products of each element and its minor's determinant, multiplied by a sign factor. This technique breaks down a larger determinant into smaller 2x2 determinants, simplifying the calculation.

Minor and Cofactor

A minor of an element in a matrix is the determinant of the smaller matrix formed by deleting the element's row and column. The cofactor is the minor multiplied by (-1)^(row+column), which accounts for the sign in expansion by minors. Understanding minors and cofactors is essential for correctly applying the expansion method.