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.
$$\left[\begin{array}{rrr} 3 & -6 & 9 \\ -2 & 7 & -2 \\ 0 & 1 & 5 \end{array}\right]$$

Key Concepts

Row Echelon Form

Row echelon form is a simplified form of a matrix achieved through a series of elementary row operations. In this form, all nonzero rows are above any rows of all zeros, and the leading coefficient (pivot) of a nonzero row is strictly to the right of the leading coefficient of the row above it, making the matrix easier to work with for solving equations or calculating determinants.

Determinant

The determinant of a square matrix is a scalar that provides useful information about the matrix, such as whether it is invertible and how it scales volumes when applied as a linear transformation. It also plays a key role in solving systems of linear equations and in various applications across linear algebra.

Elementary Row Operations

Elementary row operations, which include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another, are fundamental tools for manipulating matrices. These operations are used to transform a matrix into row echelon form or reduced row echelon form, and while some operations affect the determinant (such as multiplying a row by a scalar), others (like row addition) do not.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method to calculate the determinant of a matrix by expanding it along a chosen row or column. This technique breaks down the determinant into smaller determinants of submatrices (minors), each multiplied by a factor (the cofactor), providing a recursive approach to evaluating the determinant.

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