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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q34E (page 309)

34. If A is any skew-symmetric $4 脳 4$ matrix, then detA = 0.

Short Answer

Expert verified

Therefore, $d e t A 鈮� 0$ the given statement is not satisfied.

Step by step solution

01

Matrix Definition

Matrix is a set of numbers arranged in rows andcolumns 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 an 鈥渕 by n鈥� matrix, written 鈥� $m 脳 n$ .鈥�

02

To check whether the given condition is true or false

For example $4 脳 4$ matrix,

$A = [\begin{matrix} 1 & 1 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$

is a skew-symmetric $4 脳 4$ matrix, yet det A = 2

Therefore,

$d e t A 鈮� 0$ .

Thus, $d e t A 鈮� 0$ the given statement is not satisfied.

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Most popular questions from this chapter

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

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

$d e t [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{matrix}] = 1$

$d e t (A + B) = d e t A + d e t B$ for all $5 脳 5$ matricesA andB.

Consider two vectors $\overset{鈬赌}{v}$ and $\overset{鈬赌}{w}$ in ${鈩�}^{3}$ . Form the matrix $A = [\overset{鈬赌}{v} x \overset{鈬赌}{w} \overset{鈬赌}{v} \overset{鈬赌}{w}]$ . Express detA in terms of $|| \overset{鈬赌}{v} x \overset{鈬赌}{w} ||$ . For which choices of $\overset{鈬赌}{v}$ and $\overset{鈬赌}{w}$ is Ainvertible?

If all the columns of a square matrixAare unit vectors, then the determinant ofAmust be less than or equal to 1.

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