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Linear Algebra With Applications

Otto Bretscher

$Math Studyset Vaia Explanations$ Math

5th Edition · 442 pages

ISBN: 9780321796974

Chapter 6: Q 6.3-8E (page 306)

Question: Demonstrate the equation for a noninvertiblen × nmatrix (Theorem 6.3.3)..

Short Answer

Expert verified

Since the columns of A are linearly dependent, this means that.

So the right side of this equation is also 0.

Step by step solution

01

Matrix Definition.

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be a “m by n” matrix, written “ m × n.”

02

To demonstrate the equation.

If

Is a non-invertible matrix, then det A = 0.

Since the columns of are linearly dependent, this means that .

So the right side of this equation is also 0.

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Most popular questions from this chapter

In Exercises 5 through 40, find the matrix of the given linear transformation with respect to the given basis. If no basis is specified, use standard basis:for,

forandfor,.For the spaceof upper triangularmatrices, use the basis

Unless another basis is given. In each case, determine whetheris an isomorphism. Ifisn’t an isomorphism, find bases of the kernel and image ofand thus determine the rank of.

21. from to with respect to the basis.

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

2. $[\begin{matrix} 1 & 2 & 3 \\ 1 & 6 & 8 \\ - 2 & - 4 & 0 \end{matrix}]$

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

9. $[\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 \\ 1 & 1 & 3 & 3 & 3 \\ 1 & 1 & 1 & 4 & 4 \\ 1 & 1 & 1 & 1 & 5 \end{matrix}]$

Consider an nxn matrix A with integer entries such that A=1. Are the entries of $A^{- 1}$ necessarily integers? Explain.

Consider two distinct real numbers, a and b. We define the function

$f (t) = d e t [\begin{matrix} 1 & 1 & 1 \\ a & b & t \\ a^{2} & b^{2} & t^{2} \end{matrix}]$

a. Show that $f (t)$ is a quadratic function. What is the coefficient of $t^{2}$ ?
b. Explain why $f (a) = f (b) = 0$ . Conclude that $f (t) = k (t - a) (t - b)$ , for some constant k. Find k, using your work in part (a).
c. For which values of tis the matrix invertible?

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