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Is there any relation between the value of determinant and the invertibility of matrix in linear algebra?
yes
However, a conditional matrix A is invertible if and only if the value of its determinant | a | is not equal to 0
A ^ (- 1) = (1 / | a |) × a *, where a ^ (- 1) represents the inverse matrix of matrix A, where | a | is the determinant of matrix A, and a * is the adjoint matrix of matrix A
We have known the eigenvalues 1,2, - 1 of the matrix of order 3. Find the value of the determinant | a * a * a + 2a-4e |
This is an example in the textbook, but I can't understand it
The cube with eigenvalues of 1 plus 2 times 1 minus 4 is equal to - 1. Similarly, the other two cubes with eigenvalues of 2 plus 2 times 2 minus 4 are equal to 8, and the third eigenvalue is - 7. Therefore, the value of the determinant is - 1 times 8 and - 7 times 56
Let the determinant of the third-order square matrix a be equal to 2, then 3A *=
｜3A*｜=3^3 *｜A*｜= 3^3 * |A|^(3-1)= 27*4 =108
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