Question

Express the simultaneous equations kx+3y=5 and 4x+(5k−y)=10 in matrix form. Find the values of k for which the above equations have:

(i) For an infinite number of solutions, the determinant of matrix A is: det(A)=5k^2−k−12=0
(ii) For no solution, the determinant of matrix A must be non-zero.
A) i only
B) ii only
C) Both i and ii
D) Neither i nor ii

Answer 1

To express the simultaneous equations in matrix form, create a coefficient matrix and a constant matrix. Find the determinant of the coefficient matrix to determine the values of k for infinite solutions. If the determinant is non-zero, there is no solution.

Step-by-step explanation:

To express the simultaneous equations kx+3y=5 and 4x+(5k−y)=10 in matrix form, we can create a coefficient matrix and a constant matrix. The coefficient matrix, A, is:

[ k, 3]

[ 4, 5k-1]

and the constant matrix, B, is:

[ 5 ]

[ 10 ]

To find the values of k for which the equations have infinite solutions, we need to find the determinant of matrix A. The determinant is det(A) = 5k^2 - k - 12. By setting this determinant equal to zero and solving for k, we can determine the values of k that satisfy this condition. This means that option (i) is correct. For option (ii), we need to find the determinant of matrix A. If the determinant is non-zero, the equations will have no solution. Therefore, the correct options are i only, so the answer is A).