How to prove that
(if A is any invertible matrix) with n>2
We know that
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Note: This property holds for square matrices which are invertible.
This property of adjoint of matrices can be easily proved using property
where adj(A) is adjoint of A, det(A) is determinant of A and
is inverse of A. A here is an invertible matrix.
The matrices are said to be singular if their determinant is equal to zero. For example, if we have matrix A whose all elements in the first column are zero. Then, by one of the property of determinants, we can say that its determinant is equal to zero. Hence, A would be called as singular matrix.
If, we have a given matrix A. We can perform some column and row operations on this matrix such that value of its determinant remains unchanged. If, we multiply particular row (column) of matrix A with a constant k and then add all the element of that row (column) to the corresponding elements of another row (column) then the value of determinant remains unchanged. This is an important property regarding matrices and determinants.
If, we have any given matrix A and we multiply any row or column of matrix with constant k then determinant of modified matrix becomes k times of determinant of A. We can show this by taking example of 3 x 3 matrix A and calculating its determinant.