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Problem 1

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. \(\begin{aligned} x-3 y &=-1 \\ 4 x+3 y &=11 \end{aligned}\)

Expert verified

The given system of linear equations has one and only one solution, which is \((x,y) = (2,1)\).

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Chapter 5

Find the inverse of the matrix, if it exists. Verify your answer. $\left[\begin{array}{rrr}3 & -2 & 7 \\ -2 & 1 & 4 \\ 6 & -5 & 8\end{array}\right]$

Chapter 5

Find the inverse of the matrix, if it exists. Verify your answer. \(\left[\begin{array}{ll}4 & 2 \\ 6 & 3\end{array}\right]\)

Chapter 5

Let $$A=\left[\begin{array}{ll}1 & 2 \\\3 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{ll}2 & 1 \\\4 & 3\end{array}\right]$$ Compute \(A B\) and \(B A\) and hence deduce that matrix multiplication is, in general, not commutative.

Chapter 5

Compute the indicated products. $\left[\begin{array}{rr}-1 & 2 \\ 3 & 1\end{array}\right]\left[\begin{array}{ll}2 & 4 \\ 3 & 1\end{array}\right]$

Chapter 5

Show that the matrices are inverses of each other by showing that their product is the identity matrix \(I\). $\left[\begin{array}{rrr}2 & 4 & -2 \\ -4 & -6 & 1 \\ 3 & 5 & -1\end{array}\right]\( and \)\left[\begin{array}{rrr}\frac{1}{2} & -3 & -4 \\\ -\frac{1}{2} & 2 & 3 \\ -1 & 1 & 2\end{array}\right]$

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