| course |
course_year |
question_number |
tags |
title |
year |
Linear Mathematics |
IB |
5 |
IB |
2002 |
Linear Mathematics |
|
3.II.17F |
2002 |
Define the determinant of an $n \times n$ matrix $A$, and prove from your definition that if $A^{\prime}$ is obtained from $A$ by an elementary row operation (i.e. by adding a scalar multiple of the $i$ th row of $A$ to the $j$ th row, for some $j \neq i$ ), then $\operatorname{det} A^{\prime}=\operatorname{det} A$.
Prove also that if $X$ is a $2 n \times 2 n$ matrix of the form
$$\left(\begin{array}{ll}
A & B \\
O & C
\end{array}\right)$$
where $O$ denotes the $n \times n$ zero matrix, then $\operatorname{det} X=\operatorname{det} A$ det $C$. Explain briefly how the $2 n \times 2 n$ matrix
$$\left(\begin{array}{ll}
B & I \\
O & A
\end{array}\right)$$
can be transformed into the matrix
$$\left(\begin{array}{cc}
B & I \\
-A B & O
\end{array}\right)$$
by a sequence of elementary row operations. Hence or otherwise prove that $\operatorname{det} A B=$ $\operatorname{det} A \operatorname{det} B$.