Neco 2017 Further Mathematics Question 5

Practice essay / theory question 5 from the 2017 Neco Further Mathematics examination. This question covers Matrices, Determinants, Matrix equations, Inverse of a matrix .

Neco 2017 Further Mathematics Essay / Theory
Topics: Matrices Determinants Matrix equations Inverse of a matrix
Question 5 NECO • 2017 • FURTHER_MATHEMATICS • essay

Let \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) be a \(2\times2\) matrix with integer entries. It is given that \(\det A = 6\) and that \(A\) satisfies the matrix equation \(A^{2} - 5A + 6I = 0\), where \(I\) is the \(2\times2\) identity matrix. In addition, the trace of \(A\) is \(a+d = 5\).

Question Parts

( a )
Show that the characteristic polynomial of \(A\) is \(\lambda^{2} - 5\lambda + 6 = 0\) and deduce the eigenvalues of \(A\).
( b )
Using the equation \(A^{2} - 5A + 6I = 0\), express \(A^{-1}\) in terms of \(A\) and \(I\).
( c )
Write \(A^{-1}\) explicitly in terms of the entries \(a, b, c, d\).

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About This Question

This is Neco 2017 Further Mathematics Question 5. It is one of the essay questions from the 2017 Neco Further Mathematics examination.

The question covers Matrices, Determinants, Matrix equations, Inverse of a matrix .

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