A square matrix A is invertible if and only ifQuestion 9Select one:ย ๐ย =ย 1ย ๐๐ ย ๐๐ย ๐๐๐๐๐๐ฃ๐๐๐ข๐ย ๐๐ย ๐ด.๐ย =ย 0ย ๐๐ ย ๐๐ย ๐๐๐๐๐๐ฃ๐๐๐ข๐ย ๐๐ย ๐ด.ย ๐ย =ย 0ย ๐๐ ย ๐๐๐กย ๐๐ย ๐๐๐๐๐๐ฃ๐๐๐ข๐ย ๐๐ย ๐ด.๐ย =ย 1ย ๐๐ ย ๐๐๐กย ๐๐ย ๐๐๐๐๐๐ฃ๐๐๐ข๐ย ๐๐ย ๐ด.
Question
A square matrix A is invertible if and only ifQuestion 9Select one:ย ๐ย =ย 1ย ๐๐ ย ๐๐ย ๐๐๐๐๐๐ฃ๐๐๐ข๐ย ๐๐ย ๐ด.๐ย =ย 0ย ๐๐ ย ๐๐ย ๐๐๐๐๐๐ฃ๐๐๐ข๐ย ๐๐ย ๐ด.ย ๐ย =ย 0ย ๐๐ ย ๐๐๐กย ๐๐ย ๐๐๐๐๐๐ฃ๐๐๐ข๐ย ๐๐ย ๐ด.๐ย =ย 1ย ๐๐ ย ๐๐๐กย ๐๐ย ๐๐๐๐๐๐ฃ๐๐๐ข๐ย ๐๐ย ๐ด.
Solution
A square matrix A is invertible if and only if ๐ = 0 is not an eigenvalue of A.
Here's why:
A square matrix is invertible if and only if its determinant is not zero. The determinant of a matrix is the product of its eigenvalues. Therefore, if 0 is an eigenvalue of the matrix, the determinant will be zero, and the matrix will not be invertible. Conversely, if 0 is not an eigenvalue of the matrix, the determinant will not be zero, and the matrix will be invertible.
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