Textbook Guide Mathematics: Parallelepiped, Main Diagonal, Triangular Matrix
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If the determinant of a matrix is non zero, then it is invertible. All matrices mentioned need to be square matrices. The determinant of a (cid:1866) (cid:1866) matrix is calculated by the formula below: (cid:1856)(cid:1857)(cid:1872)(cid:1827)=(cid:2869)(cid:2869)(cid:1856)(cid:1857)(cid:1872)(cid:1827)(cid:2869)(cid:2869) (cid:2869)(cid:2870)(cid:1856)(cid:1857)(cid:1872)(cid:1827)(cid:2869)(cid:2870)+(cid:2869)(cid:2871)(cid:1856)(cid:1857)(cid:1872)(cid:1827)(cid:2869)(cid:2871) (cid:1710)+(cid:4666) (cid:883)(cid:4667)(cid:2869)+(cid:2869)(cid:1856)(cid:1857)(cid:1872)(cid:1827)(cid:2869) Here, (cid:1827)(cid:2869)(cid:3037) named matrices are the submatrices obtained by closing the 1st row and column (cid:1862) in the matrix (cid:1827) . The closed row (1st row) and column elements are eliminated and the remaining entries form (cid:1827)(cid:2869)(cid:3037) submatrices. (cid:1829)(cid:3036)(cid:3037)=(cid:4666) (cid:883)(cid:4667)(cid:3036)+(cid:3037)(cid:1856)(cid:1857)(cid:1872)(cid:1827)(cid:3036)(cid:3037) is called the cofactor of matrix (cid:1827) . So, rewriting the above (cid:1856)(cid:1857)(cid:1872)(cid:1827)=(cid:2869)(cid:2869)(cid:1829)(cid:2869)(cid:2869)+(cid:2869)(cid:2870)(cid:1829)(cid:2869)(cid:2870)+(cid:1710)+(cid:2869)(cid:1829)(cid:2869) : this is the cofactor expansion across the 1st row of matrix (cid:1827). If we want to write the cofactor expansion across the (cid:1861) row or (cid:1862) column, here (cid:1856)(cid:1857)(cid:1872)(cid:1827)=(cid:3036)(cid:2869)(cid:1829)(cid:3036)(cid:2869)+(cid:3036)(cid:2870)(cid:1829)(cid:3036)(cid:2870)+(cid:1710)+(cid:3036)(cid:1829)(cid:3036) (cid:1856)(cid:1857)(cid:1872)(cid:1827)=(cid:2869)(cid:3037)(cid:1829)(cid:2869)(cid:3037)+(cid:2870)(cid:3037)(cid:1829)(cid:2870)(cid:3037)+(cid:1710)+(cid:3037)(cid:1829)(cid:3037) The determinant of a triangular matrix is the multiplication of the entries on the main diagonal. Hint: the easiest way to calculate a determinant is by choosing the way that necessitates the least operations.
