Determinant of two matrices

**TheFangel** · May 10th 2010, 02:41 AM

Hi,

I didn't get so well what this problem is asking. Mostly how the B matrix is generated.

Let $\lambda \in R$ . The matrix B is generated by the matrix A by addition of the $\lambda$ multiple of the j-th row to the i-th row ( $i \neq j$ ). Show that detB = detA.

**HallsofIvy** · May 10th 2010, 03:34 AM

Originally Posted by TheFangel

Hi,

I didn't get so well what this problem is asking. Mostly how the B matrix is generated.

Let $\lambda \in R$ . The matrix B is generated by the matrix A by addition of the $\lambda$ multiple of the j-th row to the i-th row ( $i \neq j$ ). Show that detB = detA.

For example, if $A= \begin{pmatrix}1 & 3 & 3 \\ 4 & 5 & 1 \\ 3 & 2 & 3\end{pmatrix}$ and B "is generated by the matrix A by addition of the $\lambda$ multiple of the 3rd row the the first row" (taking j= 3 and i= 1), then $B= \begin{pmatrix}1+ 3\lambda & 3+ 2\lambda & 3+ 3\lambda \\ 4 & 5 & 1\\ 3 & 2 & 3 \end{pmatrix}$ .

A determinant is a sum of products, each of which involves exactly one factor from each row. Here, such a product, $a_{1i}a_{2j}a_{3k}...$ will be replace by $a_{1i}a_{2j}(a_{3k}+ \lambda a_{nk})...$ . Multiply that out.

**TheFangel** · May 11th 2010, 05:06 AM

Wow!
Thanks very much!

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