1. If each element of a row (column) is a zero, then `Delta =0`.
2. If two rows (columns) are proportional, then `Delta = 0`.
3. If any two rows (columns) are interchanged odd times, then `Delta` becomes `-Delta`
4. If the rows and columns are interchanged, then `Delta` is unchanged i.e. `|A^T|= |A |`.
5. If each element of a row (column) of a determinant is multiplied by a constant `k`, then the value of the new determinant is `k` times the value of the original determinant
i.e., `|(ka , kb , kc),(p, q, r),(u, v, w)|= k |(a, b, c),(p, q, r), (u, v , w)|`
6. `|(a_1+a_2 , b , c),(p_1+p_2 , q, r), (u_1+u_2 , v , w)|=|(a_1 , b, c), (p, q , r),(u_1 , v , w)| + | (a_2 , b , c), (p_2 , q , r), (u_2 , v , w)|`
7. If a scalar multiple of any row (column) is added to another row (column), then `Delta` is unchanged
i.e., `|(a, b, c) , (p, q , r), (u, v, w)| = | (a, b, c), (p+ka, q+kb , r+kc), (u, v, w)|`
8. Let `Delta(x)` be a `3rd` order determinant having polynomials as its elements.
(i) If `Delta (a)` has two rows (columns) proportional, then `(x- a)` is a factor of `Delta (x)`.
(ii) If `Delta (a)` has three rows (columns) proportional, then `(x- a )^2` is a factor of `Delta (x)`.
9. Product of two determinants
i.e., `|AB|= |A||B| =|BA| =|AB^T| =|A^T B| =|A^T B^T|`
10. If `Delta( x) =|(f_1 (x), f_2(x) , f_3(x)), (g_1(x) , g_2(x), g_3(x)), (a, b, c)|` then
(i) `sum_(x=1)^n Delta (x) = |(sum_(x=1)^n f_1(x) , sum_(x=1)^n f_2(x) , sum_(x=1)^n f_3(x)), (sum_(x=1)^n g_1(x) , sum_(x=1)^n g_2(x) , sum_(x=1)^n g_3(x)), (a, b, c)|`
(ii) `prod_(x=1)^n Delta (x) = |(prod_(x=1)^n f_1(x) , prod_(x=1)^n f_2(x) , prod_(x=1)^n f_3(x)), (prod_(x=1)^n g_1(x) , prod_(x=1)^n g_2(x) , prod_(x=1)^n g_3(x)),(a, b, c)|`
11. `det(kA)=k^n det (A) `, if `A` is of order `n xx n`.
12. `det (A^n) =(det A)^n `, if `n in I^+`
13. `|A^T| =|A|` where `A^T` is a transpose of a matrix.
`"Symmetric Determinant:"`
`|(a, h, g),(h,b,f),(g,f,c)|= abc + 2 fgh - af^2-bg^2- ch^2`
`"Skew-symmetric Determinant"`
`|(0, h, -g), (-h, 0, f), (g, -f, 0)|=0`
In skew-symmetric matrix of odd order, the value of determinant is zero.
1. If each element of a row (column) is a zero, then `Delta =0`.
2. If two rows (columns) are proportional, then `Delta = 0`.
3. If any two rows (columns) are interchanged odd times, then `Delta` becomes `-Delta`
4. If the rows and columns are interchanged, then `Delta` is unchanged i.e. `|A^T|= |A |`.
5. If each element of a row (column) of a determinant is multiplied by a constant `k`, then the value of the new determinant is `k` times the value of the original determinant
i.e., `|(ka , kb , kc),(p, q, r),(u, v, w)|= k |(a, b, c),(p, q, r), (u, v , w)|`
6. `|(a_1+a_2 , b , c),(p_1+p_2 , q, r), (u_1+u_2 , v , w)|=|(a_1 , b, c), (p, q , r),(u_1 , v , w)| + | (a_2 , b , c), (p_2 , q , r), (u_2 , v , w)|`
7. If a scalar multiple of any row (column) is added to another row (column), then `Delta` is unchanged
i.e., `|(a, b, c) , (p, q , r), (u, v, w)| = | (a, b, c), (p+ka, q+kb , r+kc), (u, v, w)|`
8. Let `Delta(x)` be a `3rd` order determinant having polynomials as its elements.
(i) If `Delta (a)` has two rows (columns) proportional, then `(x- a)` is a factor of `Delta (x)`.
(ii) If `Delta (a)` has three rows (columns) proportional, then `(x- a )^2` is a factor of `Delta (x)`.
9. Product of two determinants
i.e., `|AB|= |A||B| =|BA| =|AB^T| =|A^T B| =|A^T B^T|`
10. If `Delta( x) =|(f_1 (x), f_2(x) , f_3(x)), (g_1(x) , g_2(x), g_3(x)), (a, b, c)|` then
(i) `sum_(x=1)^n Delta (x) = |(sum_(x=1)^n f_1(x) , sum_(x=1)^n f_2(x) , sum_(x=1)^n f_3(x)), (sum_(x=1)^n g_1(x) , sum_(x=1)^n g_2(x) , sum_(x=1)^n g_3(x)), (a, b, c)|`
(ii) `prod_(x=1)^n Delta (x) = |(prod_(x=1)^n f_1(x) , prod_(x=1)^n f_2(x) , prod_(x=1)^n f_3(x)), (prod_(x=1)^n g_1(x) , prod_(x=1)^n g_2(x) , prod_(x=1)^n g_3(x)),(a, b, c)|`
11. `det(kA)=k^n det (A) `, if `A` is of order `n xx n`.
12. `det (A^n) =(det A)^n `, if `n in I^+`
13. `|A^T| =|A|` where `A^T` is a transpose of a matrix.
`"Symmetric Determinant:"`
`|(a, h, g),(h,b,f),(g,f,c)|= abc + 2 fgh - af^2-bg^2- ch^2`
`"Skew-symmetric Determinant"`
`|(0, h, -g), (-h, 0, f), (g, -f, 0)|=0`
In skew-symmetric matrix of odd order, the value of determinant is zero.