Suppose that A=[ a b c d e f g h i ] and det(A)=4. Com

step 1 All right, so we're starting from matrix A, and we're going to use elementary row operations to

Suppose that A=[ a b c d e f ...

Suppose that $$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right] \text { and } \operatorname{det}(A)=4$$. Compute the determinant of each matrix below.
$$\left[\begin{array}{rrr} g & h & i \\ -4 a & -4 b & -4 c \\ 2 d & 2 e & 2 f \end{array}\right]$$

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Key Concepts

Determinant Properties

The determinant is a scalar value that is associated with a square matrix. It is a function that has several key properties, such as being linear in each row, changing sign when two rows are interchanged, and scaling multiplicatively when a row is multiplied by a scalar. These properties allow one to compute the determinant of complex matrices by relating them to simpler forms or by applying row operations.

Scalar Multiplication of Rows

When a row in a matrix is multiplied by a scalar, the determinant of the matrix is also multiplied by that scalar. If multiple rows are multiplied by different scalars, the overall determinant is the product of the original determinant and all those scalar factors. This concept is crucial when rows are scaled, as seen in the problem where one row is multiplied by -4 and another by 2.

Row Interchanges and Permutations

Reordering the rows of a matrix affects the sign of its determinant. Specifically, each time two rows are swapped, the determinant is multiplied by -1. The overall impact of reordering rows is determined by the parity of the number of swaps involved. This principle is used to relate the determinant of a matrix with rearranged rows to the determinant of the original matrix.

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