If the nxn matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -B.
The Matrix -B is symmetric.
A square matrix is symmetric matrix if
Given that A and B are symmetric matrices and B is invertible.
Since A is symmetric, then .
Then by properties of transpose, it gives,
Therefore, -B is a symmetric matrix.
Leonardo da Vinci and the resolution of forces. Leonardo (1452–1519) asked himself how the weight of a body, supported by two strings of different length, is apportioned between the two strings.
Three forces are acting at the point D: the tensions and in the strings and the weight . Leonardo believed that
Was he right? (Source: Les Manuscripts de Léonard de Vinci, published by Ravaisson-Mollien, Paris, 1890.)
Hint: Resolve into a horizontal and a vertical component; do the same for . Since the system is at rest, the equation holds. Express the ratios
and . In terms of and , using trigonometric functions, and compare the results.
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