Determinant

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date: 2025-01-07

Determinant

Remark

If A contains a row or column of zeros the det is 0 as each term in the sum has a 0 term

Exercise

Prove that the determinant of a diagonal matrix is the product of the diagonal

if then the product has atleast one term such that where as for some hence the product is zero and the only non-zero term in the determinant is when that is

Corollary

If A is an upper triangle then is the product of diagonal entries

If then .
but as hence similarly and hence

Theorem

Proposition

Let be obtained from A by multplying a row (or colum) of A by a scalar then

(::todo)

is obtained from A by interchanging two rows (or columns) then

If

Let be obtained from A by inter changing rows and

as runs over

Proposition

If two rows and columns of A are equal than det

Suppose th and th rows of A are equal interchanging will alter the det by -1 so if in if in , is of char 2 we pair the term in the expression of det A with the term . so that the addup to 0

let the kth row of A be the sum of 2 rows and Then is obtained from A by replacing the kth row of A with and is obtained by replacing th row of A with

For a fixed

Corollary

If a scalar multiple of a row (or column ) is added to a row (or column) determinant remains unchanged

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