this is called equations utilizng inverses
how would i solve this problem:
[ 15 18 ] = [ 6 -3]
[ 34 27 ] [ 7 -2] A
To solve this problem, you need to find the matrix A. The given equation [ 15 18 ] = [ 6 -3] is in the form of a matrix equation, [ 34 27 ] [ 7 -2] A.
To find the matrix A, you need to eliminate the given matrices on the right side of the equation by using matrix inverses.
Step 1: Find the inverse of the matrix on the right side. Let's call it B.
To find the inverse of a 2x2 matrix, you need to follow these steps:
1. Swap the positions of the diagonal elements: B = [ -2 3 ]
[ -7 6 ]
2. Multiply every element by the reciprocal of the determinant of the original matrix.
The determinant of the matrix [ 7 -2 ] is (7 * 6) - ( -2 * -7) = 42 + 14 = 56.
Therefore, the reciprocal of the determinant is 1/56.
Multiply every element of B by 1/56: B = [ (-2/56) (3/56) ]
[ (-7/56) (6/56) ]
Simplifying: B = [ -1/28 3/56 ]
[ -1/8 3/28 ]
Step 2: Multiply both sides of the equation by the inverse of the matrix on the right side.
[ 15 18 ] [ -1/28 3/56 ] = [ 6 -3] [ -1/28 3/56 ]
[ 34 27 ] [ -1/8 3/28 ] [ 7 -2] [ -1/8 3/28 ] A
Simplifying this multiplication yields:
[ 15 18 ] [ -1/28 3/56 ] = [ 6*(-1/28) + (-3)*(-1/8) 6*(3/56) + (-3)*(3/28) ]
[ 34 27 ] [ -1/8 3/28 ] [ 7*(-1/28) + (-2)*(-1/8) 7*(3/56) + (-2)*(3/28) ] A
[ 15* (-1/28) + 18 * (-1/8) 15*(3/56) + 18 * (3/28) ] = [ 6*(-1/28) + (-3)*(-1/8) 6*(3/56) + (-3)*(3/28) ]
[ 34* (-1/28) + 27 * (-1/8) 34*(3/56) + 27 * (3/28) ] [ 7*(-1/28) + (-2)*(-1/8) 7*(3/56) + (-2)*(3/28) ] A
Now you have a system of linear equations. Simplify each equation.
[ -15/28 - 9/8 + 45/56 + 27/14 ] = [ -6/28 + 3/8 + 18/56 - 9/14 ]
[ -17/28 - 27/8 + 51/56 + 81/28 ] [ -7/28 + 2/8 + 21/56 - 6/14 ] A
[ (-15 - 63 + 45 + 108) / 28 ] = [ (-6 + 21 + 18 - 18) / 28 ]
[ (- 17 - 216 + 51 + 81) / 28 ] [ (-7 + 28 + 21 - 12) / 28 ] A
[ 75 / 28 ] = [ 15/28 ]
[ -101 / 28 ] [ 30/28 ] A
Step 3: Equate the corresponding elements on both sides and solve for matrix A.
75 / 28 = 15 / 28
-101 / 28 = 30 / 28
Simplifying these equations gives us:
75 = 15
-101 = 30
Since these equations are not satisfied, there is no solution for matrix A that satisfies the given equation [ 15 18 ] = [ 6 -3] [ 34 27 ] [ 7 -2] A.