CAIE A-Level Further Mathematics 2.2.6 Characteristic Equations

Practise substituting a matrix into its characteristic equation and rearranging the identity to express high powers, the inverse or an inverse power efficiently.

Exam points

replace the scalar variable by the matrix and include the identity matrix in the constant term
multiply or divide the matrix identity by a suitable power to isolate the required inverse
substitute the lower-power identity repeatedly to express a high power in the requested basis

[Maximum number: 6]

It is given that a is a positive constant.

Use the characteristic equation of A to find $\mathbf{A}^{-1}$ .

$(\lambda-a)(\lambda+1)(\lambda+4)=\lambda^{3}+(5-a) \lambda^{2}+(4-5 a) \lambda-4 a=0$

B1

Finds characteristic equation.

$4 a \mathbf{A}^{-1}=\mathbf{A}^{2}+(5-a) \mathbf{A}+(4-5 a) \mathbf{I}$

M1 A1

Multiplies through by $\mathbf{A}^{-1}$ .

$\mathbf{A}^{2}=\left(\begin{array}{ccc}a^{2} 2 a^{2}-17 a^{2}-1 0 16 0 -15 1\end{array}\right)$

B1

$4 a \mathbf{A}^{-1}=\left(\begin{array}{ccc}a^{2} 2 a^{2}-17 a^{2}-1 0 16 0 -15 1\end{array}\right)+(5-a)\left(\begin{array}{ccc}a 2 a+5 a+1 0 -4 0 3 -1\end{array}\right)+(4-5 a)\left(\begin{array}{lll}1 0 0 1 0 0 1\end{array}\right)$

M1

Substitutes for and A in correct equation.

$\mathbf{A}^{-1}=\frac{1}{4 a}\left(\begin{array}{ccc}4 5 a+8 4 a+4 0 -a 0 -3 a -4 a\end{array}\right)$

A1

6

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2.2.6—Eigen terminology question 1