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Chapter 1: Q15E (page 39)

Determine whether the statements that follow are true or false, and justify your answer.

15: The systemAx⇀=[0001]isinconsistent for all 4×3 matrices A.

Short Answer

Expert verified

True, the given system of equations is inconsistent.

Step by step solution

01

Corresponding augmented matrix

We have given that, a system of equations lets the A matrix be

aeibfjcgkdhl

Now, we have given the system.

Ax⇀=0001

02

Justification of answer

Put the value of A matrix in the above equation we will get.

aeibfjcgkdhlx⇀=0001

Now since the matrix A has 3 columns then it has a maximum of 3 pivot entry.

Also, the pivot entry not possible in the last row.

This means for any value ofx→ the value 0≠1

Hence, the given statement is true, because the given system is inconsistent.

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Most popular questions from this chapter

Question: If A is a non-zero matrix of the form,[a-bba] then the rank of A must be 2.

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)

Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a line in \({\mathbb{R}^3}\).

In Exercises 32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

32. \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&{ - 3}&9&5\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&0&0&{ - 1}\end{array}} \right]\)

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \({T_1},...,{T_4}\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes—to the left, above, to the right, and below. For instance,

\({T_1} = \left( {10 + 20 + {T_2} + {T_4}} \right)/4\), or \(4{T_1} - {T_2} - {T_4} = 30\)

33. Write a system of four equations whose solution gives estimates

for the temperatures \({T_1},...,{T_4}\).
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