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Section 3.12 Matrix Inverses: Homework
Find the inverse of each of the following matrices if it exists. If not, explain why.
\(\displaystyle \left(\begin{array}{rr} 5\amp -6\\15\amp -18 \end{array} \right)\)
\(\displaystyle \begin{pmatrix}4\amp 5\\5\amp 6 \end{pmatrix}\)
\(\displaystyle \left(\begin{array}{rrr} 2\amp 7\amp 1\\7\amp 2\amp 1 \end{array} \right)\)
\(\displaystyle \left(\begin{array}{rrr} 3\amp 0\amp -1\\7\amp 2\amp -5\\2\amp -4\amp 2 \end{array} \right)\)
\(\displaystyle \left(\begin{array}{rrr}1\amp 2\amp -3\\2\amp 5\amp -4\\-1\amp -1\amp 6 \end{array} \right)\)
\(\displaystyle \left(\begin{array}{rrr}1\amp 2\amp -1\\2\amp 5\amp 0\\-1\amp 0\amp 6 \end{array} \right)\)
\(\displaystyle \left(\begin{array}{rrr}1\amp 3\amp -2\\3\amp -6\amp 7\\11\amp -12\amp 17 \end{array} \right)\)
\(\displaystyle \left(\begin{array}{rrr}0\amp 1\amp 7\\1\amp -1\amp 2\\2\amp -2\amp 5 \end{array} \right)\)
Using the same technique, but using only row-swaps as the row operations, find the inverse of each of the following. Identify each inverse matrix as one of the matrices in this problem, labeled \(P_{ijk}\) (Note: Don’t confuse these matrices with the elmentary matrices defined in this section):
\(\displaystyle P_{312}:=\begin{pmatrix}0\amp 0\amp 1\\1\amp 0\amp 0\\0\amp 1\amp 0 \end{pmatrix}\)
\(\displaystyle P_{321}:=\begin{pmatrix}0\amp 0\amp 1\\0\amp 1\amp 0\\1\amp 0\amp 0 \end{pmatrix}\)
\(\displaystyle P_{213}:=\begin{pmatrix}0\amp 1\amp 0\\1\amp 0\amp 0\\0\amp 0\amp 1 \end{pmatrix}\)
\(\displaystyle P_{132}:=\begin{pmatrix}1\amp 0\amp 0\\0\amp 0\amp 1\\0\amp 1\amp 0 \end{pmatrix}\)
\(\displaystyle P_{231}:=\begin{pmatrix}0\amp 1\amp 0\\0\amp 0\amp 1\\1\amp 0\amp 0 \end{pmatrix}\)
Suppose \(A\in\R^{3\times 3}\) satisfies \(A\vec{x}\ne\begin{pmatrix}0\\3\\1 \end{pmatrix}\) for every \(\vec{x}\in\R^3\text{.}\) Explain what this means about the invertibility of \(A\text{.}\)
Suppose \(A\in\R^{n\times n},\,\vec{b}\in\R^n\) are such that \(A\vec{x}=\vec{b}\) has an infinite set of solutions. Explain what this means about the invertibility of \(A\text{.}\)
Suppose \(A\in\R^{3\times 3}\) and let \((x_1\,\,x_2\,\,x_3)\) be a non-zero row vector in \(\R^3\text{.}\) Suppose \(A\) satisfies \((x_1\,\,x_2\,\,x_3)A=(0\,\,0\,\,0)\text{.}\) Explain what this means about the invertibility of \(A\text{.}\)
Prove in
Theorem 3.10.25 Part 1 if and only if
Part 8 . You may use other parts of the theorem in your proof only if you have already proved them.
Prove in
Theorem 3.10.25 Part 1 if and only if
Part 6 . You may use other parts of the theorem in your proof only if you have already proved them.
Prove in
Theorem 3.10.25 Part 1 if and only if
Part 13 . You may use other parts of the theorem in your proof only if you have already proved them.
Prove in
Theorem 3.10.25 Part 1 if and only if
Part 9 . You may use other parts of the theorem in your proof only if you have already proved them.
Prove in
Theorem 3.10.25 Part 1 if and only if
Part 12 . You may use other parts of the theorem in your proof only if you have already proved them.
Prove in
Theorem 3.10.25 Part 1 if and only if
Part 3 . You may use other parts of the theorem in your proof only if you have already proved them.
Prove in
Theorem 3.10.25 Part 1 if and only if
Part 4 . You may use other parts of the theorem in your proof only if you have already proved them.
