Final answer:
To determine the value(s) of k for which ab=ba, where a=4 2 -1 2 and b=2 4 -2 k, solve the system given by equating the resulting matrices from ab and ba. The commutative property holds if both products are equal. Solve for k by setting the corresponding elements equal.
Step-by-step explanation:
To find what value(s) of k will make ab = ba, we first need to denote the two matrices a and b. If a = 4 2 -1 2 and b = 2 4 -2 k, multiplying the matrices ab and ba will result in two 2x2 matrices. The property we are seeking is known as commutative property for matrices which holds true if ab equals ba.
The product of matrix a times matrix b (denoted as ab) is calculated by taking the sum of the products of the rows of a by the columns of b. Conversely, ba is the product of matrix b times matrix a, summing the products of the rows of b with the columns of a. Set the resulting matrices equal to each other and solve for k.
We need to match the elements of the resulting matrices in both products, hence the equations for the elements would be:
4*2+2*(-1)=2*4+4*2
4*4+2*2=-2*4-k*2
-1*2+2*2=-2*2-k*4
-1*4+2*-2=2*4+2*k
The system will provide us the value or values of k that satisfy the equation ab = ba.