# Determinants of related matrices.

Given:

det $\begin{bmatrix} r & s & t \\ u & v & w \\ x & y & z \\ \end{bmatrix}=4$, compute det $\begin{bmatrix}r&s&s\\x-8r&y-8s&z-8t\\8u&8v&8w\end{bmatrix}$

How should I start going about this problem? I'm not sure how exactly the determinants of the two matrices are related.

The determinant satisfies several properties:

1. If you multiply a single row by a scalar, it multiplies the determinant by that scalar.
2. If you swap two rows, it flips the sign of the determinant.
3. If you add a multiple of one row to another row, it doesn't change the determinant.

Using these three properties, try to find a way to use those three operations to turn your first matrix, whose determinant you know, into the new matrix.

Note that row 3 was exchanged with $8 \times$ row 2:

• How does exchanging a row affect the determinant?
• How does multiplying a row by 8 affect the determininant of the matrix?

Then $-8 \times$ row 1 was added to the (exchanged) row 2.

• How does adding a multiple of a row by affect the determinant?

These are elementary row operations which transform the determinant of the original matrix.

• Good: It just takes a little "backwards thinking" to backtrack and see which row operations were performed. – Namaste Feb 26 '13 at 20:26