CAIE A-Level Further Math A2 2.2.6 Eigen Terminology Questions

Practise using characteristic equations to reduce matrix powers, find inverse matrices and derive polynomial matrix identities with Further Mathematics Paper 2 questions.

Exam points

Use characteristic equations to reduce powers of matrices and find inverse matrices.

[Maximum number: 4]

The matrix A is given by

\mathbf{A}=\left(\begin{array}{rrr} -6 & 2 & 13 \\ 0 & -2 & 5 \\ 0 & 0 & 8 \end{array}\right) .

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

$\boldsymbol{A}^{3}-52 \boldsymbol{A}-96 \mathbf{I}=0$
Substitutes A into characteristic equation.

$96 \mathbf{A}^{-1}=\mathbf{A}^{2}-52 \mathbf{I}$
Multiples through by $\mathbf{A}^{-1}$ .

$\mathbf{A}^{2}=\left(\begin{array}{ccc}36 -16 0 4 0 0 64\end{array}\right)$
7(b)

$\mathbf{A}^{-1}=\left(\begin{array}{ccc}-\frac{1}{6} -\frac{1}{6} \frac{3}{8} 0 -\frac{1}{2} \frac{5}{16} 0 0 \frac{1}{8}\end{array}\right)$
4

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CAIE A-Level Further Math A2 2.2.6 Eigen Terminology Questions question 1