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Linear Algebra With Applications

Otto Bretscher

$Math Studyset Vaia Explanations$ Math

5th Edition · 442 pages

ISBN: 9780321796974

Chapter 6: Q 6.2-59E (page 293)

Question: If the equationdetA= detBholds for two n × n matrices Aand B, isA necessarily similar toB ?

Short Answer

Expert verified

Therefore, A andB are not similar.

Step by step solution

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.”

To find if A is similar to B

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Most popular questions from this chapter

Consider the function $F (A) = F [\vec{v} \vec{w}] = \vec{v} \times \vec{w}$ from $ℝ^{2 \times 2}$ to $ℝ$ , the dot product of the column vectors of A.
a. Is Flinear in both columns of A? See Example 6.
b. Is F linear in both rows of A?
c. Is Falternating on the columns of A? See Example 4.

(For those who have studied multivariable calculus.) Let Tbe an invertible linear transformation from $ℝ^{2}$ to $ℝ^{2}$ , represented by the matrix M. Let $Ω_{1}$ be the unit square in $ℝ^{2}$ and $Ω_{2}$ its image under T . Consider a continuous function $f (x, y)$ from $ℝ^{2}$ to $ℝ$ , and define the function $g (u, v) = f (T (u, v))$ . What is the relationship between the following two double integrals?

$\iint_{Ω 2} f (x, y) dA$ and $\iint_{Ω 1} g (u, v) dA$

Your answer will involve the matrix M. Hint: What happens when $f (x, y) = 1$ , for all $x, y$ ?

If all the entries of a square matrixAare integers and detA=1 , then the entries of matrix $A^{- 1}$ must be integers as well.

If an $n \times n$ matrixAis invertible, then there must be an $(n - 1) \times (n - 1)$ sub matrix of(obtained by deleting a row and a column of) that is invertible as well.

Find all $2 \times 2$ matrices Asuch that $adj (A) = A^{T}$ .

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Question: If the equationdetA= detBholds for two n × n matrices Aand B, isA necessarily similar toB ?

Short Answer

Step by step solution

Matrix Definition

To find if A is similar to B

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Most popular questions from this chapter

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