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Problem 10

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\frac{5}{4} x-\frac{2}{3} y=3\) \(\frac{1}{4} x+\frac{5}{3} y=6\)

Expert verified

The system of linear equations has one unique solution, which is \(x=4\) and \(y=2\).

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Chapter 5

Let $$A=\left[\begin{array}{rr}2 & 3 \\\\-4 & -5\end{array}\right]$$ a. Find \(A^{-1}\). b. Show that \(\left(A^{-1}\right)^{-1}=A\).

Chapter 5

Let $$A=\left[\begin{array}{rr}2 & 4 \\\5 & -6\end{array}\right] \text { and } B=\left[\begin{array}{rr}4 & 8 \\ -7 & 3\end{array}\right]$$ a. Find \(A^{T}\) and show that \(\left(A^{T}\right)^{T}=A\). b. Show that \((A+B)^{T}=A^{T}+B^{T}\). c. Show that \((A B)^{T}=B^{T} A^{T}\).

Chapter 5

Compute the indicated products. $\left[\begin{array}{rr}1 & 3 \\ -1 & 2\end{array}\right]\left[\begin{array}{lll}1 & 3 & 0 \\ 3 & 0 & 2\end{array}\right]$

Chapter 5

Find the matrix \(A\) such that $$A\left[\begin{array}{rr}1 & 0 \\\\-1 & 3\end{array}\right]=\left[\begin{array}{rr}-1 & -3 \\\3 & 6\end{array}\right]$$

Chapter 5

Show that the matrices are inverses of each other by showing that their product is the identity matrix \(I\). \(\left[\begin{array}{ll}4 & 5 \\ 2 & 3\end{array}\right]\) and $\left[\begin{array}{rr}\frac{3}{2} & -\frac{5}{2} \\ -1 & 2\end{array}\right]$

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