Let A = 1 2 3 4 5 6 3 6 9 . Find determinant value of A .

Given, A = 1 2 3 4 5 6 3 6 9 Determinant of A = 1 2 3 4 5 6 3 6 9 = 1 45 - 36 - 2 36 - 18 + 3 24 - 15 = 9 - 36 + 27 = 0

If A is 4 × 4 matrix and | A | = 3 , then find determinant of a d j A .

Given, Order of matrix A is 4 × 4 and A = 3 Using properties of adjoint of matrix, a d j A = A n - 1 = 3 4 - 1 = 3 3 = 27

If A is 3 × 3 matrix and | A | = 4 , then find | adj ( adj A ) | .

Given, Order of matrix A is 3 × 3 and A = 4 a d j a d j A = ? Using properties of adjoint of a matrix, a d j a d j A = A n - 1 2 = 4 3 - 1 2 = 4 4 = 4 × 4 × 4 × 4 = 256

Find the value of 4 3 1 0 - 1 2 0 3 1 .

4 3 1 0 - 1 2 0 3 1 = 4 - 1 × 1 - 3 × 2 - 3 0 × 1 - 0 × 2 + 1 0 × 3 - 1 × 0 = 4 - 1 - 6 - 0 - 0 = - 7 × 4 = - 28

Find the value of x if x 2 2 x = 12 .

Given, x 2 2 x = 12 Solving the determinant we get, x × x - 2 × 2 = 12 x 2 - 4 = 12 x 2 = 16 x = ± 16 x = ± 4

Find the area of the triangle whose vertices are ( 3 , 8 ) , ( - 4 , 2 ) and ( 5 , 1 ) .

The area of triangle with vertices ( 3 , 8 ) , ( - 4 , 2 ) , 5 , 1 is given as follows Δ = 1 2 3 8 1 − 4 2 1 5 1 1 Expanding the determinant along C 1 , we have = 1 2 [ 3 ( 2 - 1 ) - 8 ( - 4 - 5 ) + 1 ( - 4 - 10 ) ] = 1 2 ( 3 + 72 - 14 ) = 61 2