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Linear Algebra With Applications

Otto Bretscher

$Math Studyset Vaia Explanations$ Math

5th Edition · 442 pages

ISBN: 9780321796974

Chapter 1: Q46E (page 40)

Question:A lower triangular 3x3 matrix has rank 3 if (and only if) the product of its diagonal entries is nonzero.

Short Answer

Expert verified

The statement, “A lower triangular 3x3 matrix has rank3. if (and only if) the product of its diagonal entries is nonzero.” is true.

Step by step solution

01

Consider the condition

The diagonal elements in triangular matrices represent the number of solutions. In turn, the number of in the triangular matrix represents the rank of a matrix.

The product of diagonal entries will be nonzero if and only if the diagonal elements are nonzero.

02

Conclusion

The product of the diagonal entries is nonzero of a lower triangular 3X3 matrix with rank 3.

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Most popular questions from this chapter

The reduced row-echelon forms of the augmented matrices of three systems are given here. How many solutions does each system have?

$a . [\begin{matrix} 1 & 0 & 2 & | \\ 0 & 1 & 3 & | \\ 0 & 0 & 0 & | \end{matrix} \begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] b . [\begin{matrix} 1 & 0 & | & 5 \\ 0 & 1 & | & 6 \end{matrix}] c . [\begin{matrix} 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & 3 \end{matrix}]$

(Compute the dot products in Exercises 10 through 12

if the products are defined)

10.

$[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}], [\begin{matrix} 1 \\ - 2 \\ 1 \end{matrix}]$

Consider some particles in the plane with position vectors $\vec{r_{1}}, \vec{r_{2}}, . ...., \vec{r_{n}}$ and masses $m_{1}, m_{2}, . ...., m_{n}$ .

The position vector of the center of mass of this system is

$\vec{r_{c m}} = \frac{1}{M} (m_{1} \vec{r_{1}} + m_{2} \vec{r_{2}} + . .... + m_{n} \vec{r_{n}})$

where $M = m_{1} + m_{2} + . .... + m_{n}$ .

Consider the triangular plate shown in the accompanying sketch. How must a total mass of be distributed among the three vertices of the plate so that the plate can be supported at the point $[\begin{matrix} 2 \\ 2 \end{matrix}]$ ; that is, $\vec{r_{c m}} = [\begin{matrix} 2 \\ 2 \end{matrix}]$ ? Assume that the mass of the plate itself is negligible.

If you sell two cows and five sheep and you buy $13$ pigs, you gain $1000$ coins. If you sell three cows and three pigs and buy nine sheep, you break even. If you sell six sheep and eight pigs and you buy five cows, you lose $600$ coins. What is the price of a cow, a sheep, and a pig, respectively? (Nine Chapters, Chapter 8, Problem 8)

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

20:Vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is a linear combination of $[\begin{matrix} 4 \\ 5 \\ 6 \end{matrix}]$ and $[\begin{matrix} 7 \\ 8 \\ 9 \end{matrix}]$ .

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