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Mathematics LibreTexts

8.3: Review Problems

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}_{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

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.