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Worksheet 3.11.1 Worksheet
A4 US

Find the inverse of the elementary matrix \(E_{(-2)21} \in \R^{3 \times 3}\) using the method of augmenting \(E_{(-2)21}\) by the identity matrix and performing elimination (as in Example 3.10.10). Check that your result is consistent with Theorem 3.7.7.
Find the inverse or explain carefully why it does not exist: \(\left(\begin{array}{rr} 3\amp 8\\1\amp 3 \end{array} \right)\)
Find the inverse or explain carefully why it does not exist: \(\left(\begin{array}{rr} 2\amp 7\\-3\amp -5 \end{array} \right)\)
Find the inverse or explain carefully why it does not exist: \(\left(\begin{array}{rr} 5\amp 2\\-10\amp -4 \end{array} \right)\)
Find the inverse or explain carefully why it does not exist: \(\left(\begin{array}{rrr} 1\amp 1\amp -1\\2\amp 3\amp -1\\0\amp -2\amp -2 \end{array} \right)\)
Find the inverse or explain carefully why it does not exist: \(\left(\begin{array}{rrr} 1\amp 1\amp -1\\2\amp 3\amp -1\\1\amp 2\amp 1 \end{array} \right)\)
Find the inverse or explain carefully why it does not exist: \(\left(\begin{array}{rrr} 3\amp 0\amp -1\\7\amp 2\amp -5\\2\amp -4\amp 2 \end{array} \right)\)
Find the inverse or explain carefully why it does not exist: \(\left(\begin{array}{rrr} 0\amp 0\amp 1\\2\amp 1\amp 5\\1\amp 1\amp 6 \end{array} \right)\)
For a \(2\times 2\) matrix \(A=\begin{pmatrix}a\amp b\\c\amp d \end{pmatrix}\) to equal its own inverse, what equations must \(a,b,c,d\) satisfy?
For each part find a \(2\times 2\) matrix \(\begin{pmatrix}a\amp b\\c\amp d \end{pmatrix}\) which is its own inverse, having
1. Two zeros
2. No zeros
Why can’t one find such a matrix with exactly three zeros?