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.
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Value of Determinant remains unchanged if we add equal multiples of all the elements of row (column) to corresponding elements of another row (column)
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.
Determinant of Matrix becomes k times by multiplying any row or column by k
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.
Determinant of a matrix changes sign if we interchange any two rows or columns present in a matrix
Determinant of a matrix changes its sign if we interchange any two rows or columns present in a matrix. We can prove this property by taking an example. We take matrix A and we calculate its determinant (|A|).
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