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Problem 10

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{ll|l}1 & 1 & 3 \\ 0 & 0 & 0\end{array}\right]\)

Expert verified

The given matrix \(\left[\begin{array}{ll|l}1 & 1 & 3 \\ 0 & 0 & 0\end{array}\right]\) satisfies all three conditions for row-reduced form: zero rows are at the bottom, pivots are to the right of those in rows above if applicable, and other elements in pivot columns are zero. Therefore, it is in row-reduced form.

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Chapter 5

Lawnco produces three grades of commercial fertilizers. A \(100-\mathrm{lb}\) bag of grade \(\mathrm{A}\) fertilizer contains \(18 \mathrm{lb}\) of nitrogen, $4 \mathrm{lb}\( of phosphate, and \)5 \mathrm{lb}$ of potassium. A \(100-\mathrm{lb}\) bag of grade \(\mathrm{B}\) fertilizer contains $20 \mathrm{lb}\( of nitrogen and \)4 \mathrm{lb}$ each of phosphate and potassium. A 100-lb bag of grade \(\mathrm{C}\) fertilizer contains \(24 \mathrm{lb}\) of nitrogen, \(3 \mathrm{lb}\) of phosphate, and \(6 \mathrm{lb}\) of potassium. How many 100 -lb bags of each of the three grades of fertilizers should Lawnco produce if a. \(26,400 \mathrm{lb}\) of nitrogen, \(4900 \mathrm{lb}\) of phosphate, and \(6200 \mathrm{lb}\) of potassium are available and all the nutrients are used? b. \(21,800 \mathrm{lb}\) of nitrogen, \(4200 \mathrm{lb}\) of phosphate, and \(5300 \mathrm{lb}\) of potassium are available and all the nutrients are used?

Chapter 5

(a) write each system of equations as a matrix equation and (b) solve the system of equations by using the inverse of the coefficient matrix. \(\quad x+2 y=b_{1}$$2 x-y=b_{2}\) where (i) \(b_{1}=14, b_{2}=5\) and (ii) \(b_{1}=4, b_{2}=-1\)

Chapter 5

Let $$A=\left[\begin{array}{rr}2 & 2 \\\\-2 & -2\end{array}\right]$$ Show that \(A^{2}=0\). Compare this with the equation \(a^{2}=0\), where \(a\) is a real number.

Chapter 5

Compute the indicated products. $\left[\begin{array}{rr}-1 & 3 \\ 5 & 0\end{array}\right]\left[\begin{array}{l}7 \\ 2\end{array}\right]$

Chapter 5

Compute the indicated products. $\left[\begin{array}{rrr}6 & -3 & 0 \\ -2 & 1 & -8 \\ 4 & -4 & 9\end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

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