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Linear Algebra With Applications
Found in: Page 289
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

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


a. Show that f(t) is a quadratic function. What is the coefficient of t2? 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 t is the matrix invertible?

Therefore, the determinant of the given matrix is given by,

a). The coefficient of is t2 is -a+b.

b) .ft=kt-at-b

C) The matrix invertible value of t is ta,b.

See the step by step solution

Step by Step Solution

Step 1: What is the coefficient of t2

a) Show that ft is a quadratic function.


Thus, f is a quadratic function, the coefficient of t2 being localid="1659512383139" -a+b .

Step 2: Explain why f(a)=f(b)=0

b) We compute:

fa=-a+ba2+a2-b2a+ab2-a2b=-a3+a2b+a3-ab2+ab2-a2b=0fa=-a+bb2+a2-b2b+ab2-a2b=-a3+a2b+a3-ab2+ab2-a2b=0Thus, fa=fb=0Consider k. We compute,kt-at-b=kt2+k-a-bt+kabFor k=-a+b, we have exactly ft=kt-at-b

Step 3: For which values of t is the matrix invertible.

We solve,


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