
8.3: Review Problems

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

1.    Let $$M=\begin{pmatrix} m^{1}_{1} & m^{1}_{2} & m^{1}_{3}\\ m^{2}_{1} & m^{2}_{2} & m^{2}_{3}\\ m^{3}_{1} & m^{3}_{2} & m^{3}_{3}\\ \end{pmatrix}$$.  Use row operations to put $$M$$ into $$\textit{row echelon form}$$.  For simplicity, assume that $$m_{1}^{1}\neq 0 \neq m^{1}_{1}m^{2}_{2}-m^{2}_{1}m^{1}_{2}$$.

Prove that $$M$$ is non-singular if and only if:
$m^{1}_{1}m^{2}_{2}m^{3}_{3} - m^{1}_{1}m^{2}_{3}m^{3}_{2} + m^{1}_{2}m^{2}_{3}m^{3}_{1} - m^{1}_{2}m^{2}_{1}m^{3}_{3} + m^{1}_{3}m^{2}_{1}m^{3}_{2} - m^{1}_{3}m^{2}_{2}m^{3}_{1} \neq 0$

2.
a)    What does the matrix $$E^{1}_{2}=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$ do to $$M=\begin{pmatrix} a & b \\ d & c \end{pmatrix}$$ under left multiplication?  What about right multiplication?
b)    Find elementary matrices $$R^{1}(\lambda)$$ and $$R^{2}(\lambda)$$ that respectively multiply rows $$1$$ and $$2$$ of $$M$$ by $$\lambda$$ but otherwise leave $$M$$ the same under left multiplication.
c)    Find a matrix $$S^{1}_{2}(\lambda)$$ that adds a multiple $$\lambda$$ of row $$2$$ to row $$1$$ under left multiplication.

3.    Let $$M$$ be a matrix and $$S^{i}_{j}M$$ the same matrix with rows $$i$$ and $$j$$ switched.  Explain every line of the series of equations proving that $$\det M = -\det (S^{i}_{j}M)$$.

4.    Let $$M'$$ be the matrix obtained from $$M$$ by swapping two columns $$i$$ and $$j$$.  Show that $$\det M'=-\det M$$.

5.    The scalar triple product of three vectors $$u,v,w$$ from $$\Re^{3}$$ is $$u\cdot(v\times w)$$.  Show that this product is the same as the determinant of the matrix whose columns are $$u,v,w$$ (in that order).  What happens to the scalar triple product when the factors are permuted?

6.    Show that if $$M$$ is a $$3\times 3$$ matrix whose third row is a sum of multiples of the other rows ($$R_{3}=aR_{2}+bR_{1}$$) then $$\det M=0$$.  Show that the same is true if one of the columns is a sum of multiples of the others.