Answer:
k = -6
w = 9 or -9 (see explanation below)
f = -7
Explanation:
The entities on the left and right of the = sign are matrices
Matrix elements are referred to using row and column numbers as indexes
The size of the matrix is number of rows x number of columns
In this problem, both matrices are of size 2 x 2 = 4
The first entry of the matrix on the left ie the top left is referred to as entry 11 where 1 is the row and 1 is also the column
Using this notation, the matrix entries for the matrix on the LHS are
a₁₁ = -12 (first row, first column)
a₁₂ = - w² (first row, second column)
a₂₁ = 2f (second row, first column)
a₂₂ = 3 (second row, second column)
These values should correspond to the exact same elements for the matrix on the RHS which has elements
a₁₁ = 2k
a₁₂ = - 81
a₂₁ = -14
a₂₂ = 3
Equating the elements in LHS matrix to the corresponding RHS matrix elements gives us
-12 = 2k ⇒ k = -6
- w² = -81
⇒ w² = 81 (by dividing both sides by -1)
⇒ w = ±9
( we can't be sure whether it is +9 or -9 since 9² = (-9)² = 81
2f = -14
⇒ f = -7 (by dividing both sides by 2)