Let A=[aij]=[log5128log58log45log425]. If Aij is the cofactor of aij, Cij=∑k=12aikAjk,1≤i,j≤2, and C=[Cij], then 8∣C∣ is equal to:
Get the complete, step-by-step math solution for: "Let {A}=a_{ij}= {cc} _5 128 & _4 5 \\ _5 8 & _4 25 . If {A}_{ij} is the cofactor of {a}_{ij}, {C}_{ij}= _{k=1}² {a}_{ik} {A}_{jk}, 1 ≤ i, j ≤ 2, and {...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Simplify the elements of matrix A
First, we simplify the elements of matrix A using logarithm properties. We express the numbers as powers of their bases and apply the change of base rule for logarithms, logbman=mnlogba. Specifically, log5128=log527=7log52, log45=log225=21log25, log58=log523=3log52, and log425=log2252=22log25=log25.
Step 2: Calculate the determinant of A
Next, we calculate the determinant of matrix A. For a 2×2 matrix [acbd], the determinant is ad-bc. We use the property logba⋅logab=1. This simplifies the determinant calculation significantly.
Step 3: Relate Cij to the adjoint matrix
The definition of Cij is given as ∑k=12aikAjk. This sum represents the element in the i -th row and j -th column of the product of matrix A and the transpose of its adjoint matrix, adj(A)T. However, a more direct interpretation is that if i=j, this sum is the determinant of A. If i=j, this sum is 0.
Step 4: Determine the elements of matrix C
Based on the property of determinants, the sum ∑k=12aikAjk is equal to ∣A∣ if i=j (diagonal elements) and 0 if i=j (off-diagonal elements). This means matrix C is a scalar multiple of the identity matrix, where the scalar is ∣A∣.
Step 5: Calculate the determinant of C
The determinant of a scalar multiple of an identity matrix is the scalar raised to the power of the matrix dimension. For a 2×2 matrix, ∣kI∣=k2. Therefore, ∣C∣=∣A∣2. We substitute the value of ∣A∣ we calculated earlier.
Step 6: Calculate 8|C|
Finally, we multiply the determinant of C by 8 to get the required value.