Evaluate the determinant of the matrix by first reducing the...

Evaluate the determinant of the matrix by first reducing the matrix to row echelon form and then using some combination of row operations and cofactor expansion.
$$\left[\begin{array}{rrr} 3 & 6 & -9 \\ 0 & 0 & -2 \\ -2 & 1 & 5 \end{array}\right]$$

Key Concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method for calculating the determinant of a matrix. It involves breaking the determinant down into smaller parts by expanding along a row or a column, using the minors (determinants of submatrices) weighted by alternating signs. This technique is particularly useful once the matrix has been simplified, such as after converting it into row echelon form.

Elementary Row Operations

Elementary row operations, which include row swapping, multiplying a row by a nonzero constant, and adding a multiple of one row to another, are used to simplify a matrix. It is crucial to understand how each type of operation affects the determinant: swapping rows changes its sign; multiplying a row by a scalar multiplies the determinant by that scalar; and adding a multiple of one row to another does not change the determinant.

Determinant

The 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 the matrix is invertible (a nonzero determinant means the matrix is invertible) and the scaling factor of the linear transformation described by the matrix.

Row Echelon Form

Row echelon form is a type of matrix form in which all nonzero rows are above any zero rows, and the leading coefficient (pivot) of each nonzero row is to the right of the leading coefficient of the row above it. Reducing a matrix to this form simplifies many computations, including solving systems of linear equations and evaluating determinants.

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