if (-2 1 x+2 )
(3 x-4 5 ) =25
(0 1 3 )
a) find the value of x
b) hence ,find (-2 1 x+2 )^-1
( 3 x-4 5 )
(o 1 3 )
To find the value of x, we need to solve the equation (matrix equation) given.
First, let's multiply the two matrices on the left-hand side:
((-2)(3) + (1)(x+2) + (x+2)(5)) = 25
((3)(0) + (x-4)(1) + (5)(3)) = 25
((0)(0) + (1)(1) + (3)(3)) = 25
Now, simplify the equation:
-6 + x + 2 + 5x + 10 = 25
3x + 6 = 25
3x = 19
x = 19/3
So, the value of x is 19/3.
Now, let's move on to part b) and find the inverse of the given matrix:
To find the inverse of a matrix, let's use the formula:
A^(-1) = (1/det(A)) * adj(A)
where det(A) is the determinant of matrix A, and adj(A) is the adjugate (transpose of the cofactor matrix).
First, calculate the determinant of the matrix:
A = (-2 1 x+2 )
( 3 x-4 5 )
(0 1 3 )
det(A) = (-2)((x-4)(3) - (1)(5)) - (1)((3)(3) - (x-4)(0)) + (x+2)((0)(1) - (3)(1))
det(A) = (-6x + 24 - 5) - (9 - 0) + (0 - 3x - 6)
det(A) = -6x + 19 - 9 - 3x - 6
det(A) = -9x + 4
Next, calculate the adjugate of the matrix:
adj(A) = (x-4 -1 2)
(5 3 -3)
(-1 -1 -2)
Now, we can find the inverse of the matrix:
A^(-1) = (1/(-9x + 4)) * (adj(A))
So, the inverse of the matrix is:
(-1(x-4)/(-9x + 4) -1/(-9x + 4) 2/(-9x + 4) )
(5/(-9x + 4) 3/(-9x + 4) -3/(-9x + 4) )
(-1/(-9x + 4) -1/(-9x + 4) -2/(-9x + 4) )