Let A be a square matrix, then the determinant formed by the elements of A without changing their respective positions is called the determinant of A and is denoted by det `A` or `I AI.`
i.e. If `A =[(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)] \ \ \ \ A = |(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)|`
`text(Properties of the Determinant of a Matrix :)`
If A and B are square matrices of same order, then
(i) `|A|` exists <==> `A` is a square matrix.
(ii) `| A'| =| A|`
(iii) `|AB| = | A| | B| `and `|AB| = | BA|`
(iv) If A is orthogonal matrix, then `|A| = ± 1`
(v) If A is skew-symmetric matrix of odd order, then `|A| = 0`
(vi) If A is skew-symmetric matrix of even order, then `| A|` is a perfect square.
(vii) `|kA| = k^n |A|,` where `n` is order of `A` and `k` is scalar.
(viii) `| A^n |= | A|^n,` where `n in N`
(ix) If `A = diag( a_1, a_2 , a_3, ... , a_n),` then `|A| = a_1 · a_2 · a_3... a_n.`
Let A be a square matrix, then the determinant formed by the elements of A without changing their respective positions is called the determinant of A and is denoted by det `A` or `I AI.`
i.e. If `A =[(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)] \ \ \ \ A = |(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)|`
`text(Properties of the Determinant of a Matrix :)`
If A and B are square matrices of same order, then
(i) `|A|` exists <==> `A` is a square matrix.
(ii) `| A'| =| A|`
(iii) `|AB| = | A| | B| `and `|AB| = | BA|`
(iv) If A is orthogonal matrix, then `|A| = ± 1`
(v) If A is skew-symmetric matrix of odd order, then `|A| = 0`
(vi) If A is skew-symmetric matrix of even order, then `| A|` is a perfect square.
(vii) `|kA| = k^n |A|,` where `n` is order of `A` and `k` is scalar.
(viii) `| A^n |= | A|^n,` where `n in N`
(ix) If `A = diag( a_1, a_2 , a_3, ... , a_n),` then `|A| = a_1 · a_2 · a_3... a_n.`