Matrix A is invertible if and only if det(A) 6= 0 . Proof. If A is invertible, then it is a product of elementary matrices. Then, by the mentioned property, the determinant of A is product of determinants of these matrices. Each of these determinants is nonzero as it must be ¡1, r 6= 0 or 1. Therefore det(A) 6= 0. Multiplying all the elements of a row or a column by a real number is the same as multiplying the result of the determinant by that number. Example. We are going to find the determinant of a 2×2 matrix to demonstrate this property of the determinants: Now we evaluate the same determinant and multiply all the entries of a row by 2.
The sum of two skew-symmetric matrices is skew-symmetric. A scalar multiple of a skew-symmetric matrix is skew-symmetric. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero. , i.e. the nonzero eigenvalues of a skew-symmetric matrix are non-real.
3. the minors; these are the determinants of the matrix with the row and column of the entry taken out; here dots are used to show those. For example, here are the minors for the first row:, , , Here is the determinant of the matrix by expanding along the first row: - + - The product of a sign and a minor is called a cofactor.

0 1 . 1 The determinant of a permutation matrix. P is 1 or −1. 1 = −1. 0 depending on whether. P exchanges an even or odd number of rows. From these three properties we can deduce many others: 4. If two rows of a matrix are equal, its determinant is zero.

More Answers (1) 0. One of the fastest ways to determine the determinant of a matrix is doing row operation. For an invertible matrix we know that row operations finally reach to identity matrix which has determinant equal to 1. For calculating determinant we can write an efficient and of course fast code to do row operation and it is not 8t6INs.
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