differential geometry w notes from teacher_Part_31

differential geometry w notes from teacher_Part_31 - 61...

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3.1. EXTERIOR ALGEBRA 61 It is easy to see that there holds also δ i 1 ... i n - p l 1 ... l n - p ε j 1 ... j p l 1 ... l n - p = ( n - p )! ε j 1 ... j p i 1 ... i n - p The set of all n × n real matrices is denoted by Mat( n , R ). The determinant is a map det : Mat( n , R ) R that assigns to each matrix A = ( A i j ) a real number det A defined by det A = X ϕ S n sign ( ϕ ) A 1 ϕ (1) ··· A n ϕ ( n ) , The most important properties of the determinant are listed below: Theorem 3.1.4 1. The determinant of the product of matrices is equal to the product of the determinants: det( AB ) = det A det B . 2. The determinants of a matrix A and of its transpose A T are equal: det A = det A T . 3. The determinant of the inverse A - 1 of an invertible matrix A is equal to the inverse of the determinant of A: det A - 1 = (det A ) - 1 4. A matrix is invertible if and only if its determinant is non-zero. The determinant of a matrix
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differential geometry w notes from teacher_Part_31 - 61...

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