Show that `|{:(ma_(1),b_(1),nc_(1)),(ma_(2),b_(2),nc_(2)),(ma_(3),b_(3),nc_(3)):}|=-mn|{:(c_(1),b_(1),a_(1)),(c_(2),b_(2),a_(2)),(c_(3),b_(3),a_(3)):}|`

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(1),c_(2)),(b_(x3,c_(3)):}|+b_(1)|{:(a_(2),c_(2)),(a_(2),c_(2)):}|+c_(1)|{:(b_(x),c_(2)),(b_(2),c_(2)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column The value of for which determinant |{:(2,3,-1),(-1,-2,k),(1,-4,1):}| vanishes, is

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(1),c_(2)),(b_(x3,c_(3)):}|+b_(1)|{:(a_(2),c_(2)),(a_(2),c_(2)):}|+c_(1)|{:(b_(x),c_(2)),(b_(2),c_(2)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column The value of the determinant |{:(2,3,4),(6,5,7),(1,-3,2):}|is:

In algebra, the determinant is useful value that can be computer from the elements of a square matrix. The determinant is represented as det 'A' or |A| and its value can be evaluated by the expansion of the determinant as given below (A) Expansion of two order determinant : (B) Expansion of 3^(rd) order determinant (i) With respect to first fow : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=a_(1)|{:(b_(1),c_(2)),(b_(x3,c_(3)):}|+b_(1)|{:(a_(2),c_(2)),(a_(2),c_(2)):}|+c_(1)|{:(b_(x),c_(2)),(b_(2),c_(2)):}| =a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))+c_(1)(a_(2)b_(3)-b_(2)a_(3)) (ii) With respect to second column : |A|=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=-b_(1)|{:(a_(2),c_(1)),(a_(3),c_(3)):}|+b_(2)|{:(a_(1),c_(1)),(a_(3),c_(3)):}|-b_(3)|{:(a_(1),c_(1)),(a_(2),c_(2)):}| Similarly a determinant can be expanded with respect to any row or column. The vaue of the determinant |{:(5,1),(3,2):}|is:

Show that |[a_(1),b_(1),-c_(1)],[-a_(2),-b_(2),c_(2)],[a_(3),b_(3),-c_(3)]|=|[a_(1),b_(1),c_(1)],[a_(2),b_(2),c_(2)],[a_(3),b_(3),c_(3)]|

if quad /_=[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

if x,y and z are not all zero and connected by the equations a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0 and (p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0 show that lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|

if Delta=det[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

Cosnsider the system of equation a_(1)x+b_(1)y+c_(1)z=0, a_(2)x+b_(2)y+c_(2)z=0, a_(3)x+b_(3)y+c_(3)z=0 if |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0 , then the system has

suppose D= |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}| and Dprime= |{:(a_(1)+pb_(1),,b_(1)+qc_(1),,c_(1)+ra_(1)),(a_(2)+pb_(2),,b_(2)+qc_(2),,c_(2)+ra_(2)),(a_(3)+pb_(3),,b_(3)+qc_(3),,c_(3)+ra_(3)):}| . Then