Row operations

The effect of transpose in determinant implies that properties 1-3 [also 4-5] of column operations preserve for row operations:

  1. $ \det B = \det A$, if $ B$ is produced by adding a multiple of the $ i$th row to the $ j$th row of $ A$.
  2. $ \det B = - \det A$, if $ B$ interchanges the $ k$th and $ k'$th rows of $ A$.
  3. $ \det B = c\cdot\det A$, if $ B$ is produced by multiplying the $ j$th row of $ A$ by $ c$.

EXAMPLES 3. Find an LU factorization of $ A$, and then compute $ det A$ using the upper triangular matrix $ U$ in each of the following.

  1. $ A = \begin{bmatrix}
1 &-4 & 2 \\
-2 & 8 &-9 \\
-1 & 7 & 0
\end{bmatrix}$
  2. $ A = \begin{bmatrix}
2 &-8 & 6 & 8 \\
3 &-9 & 5 &10 \\
-3 & 0 & 1 &-2 \\
1 &-4 & 0 & 6
\end{bmatrix}$



Department of Mathematics
Last modified: 2005-10-21