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

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

If x 2 + 3 x x - 1 x + 3 x + 1 1 - 2 x x - 4 x - 2 x + 4 3 x = A x 4 + B x 3 + C x 2 + D x + E . Find E .

Given, x 2 + 3 x x - 1 x + 3 x + 1 1 - 2 x x - 4 x - 2 x + 4 3 x = A x 4 + B x 3 + C x 2 + D x + E … … 1 Now, x 2 + 3 x x - 1 x + 3 x + 1 1 - 2 x x - 4 x - 2 x + 4 3 x = x 2 + 3 x 3 x 1 - 2 x - x + 4 x - 4 - x - 1 3 x x + 1 - x - 2 x - 4 + x + 3 x + 1 x + 4 - x - 2 1 - 2 x = x 2 + 3 x 3 x - 6 x 2 - x 2 + 16 - x - 1 3 x 2 + 3 x - x 2 + 6 x - 8 + x + 3 x 2 + 5 x + 4 - x + 2 + 2 x 2 - 4 x = x 2 + 3 x - 7 x 2 + 3 x + 16 - x - 1 2 x 2 + 9 x - 8 + x + 3 3 x 2 + 6 = - 7 x 4 - 21 x 3 + 3 x 3 + 9 x 2 + 16 x 2 + 48 x - 2 x 3 + 7 x 2 - 17 x + 8 + 3 x 3 + 9 x 2 + 6 x + 18 = - 7 x 4 - 21 x 3 + 3 x 3 + 9 x 2 + 16 x 2 + 48 x - 2 x 3 + 7 x 2 - 17 x + 8 + 3 x 3 + 9 x 2 + 6 x + 18 Comparing it with equation 1 we get, E = 18 - 8 E = 10