Hint
\(\displaystyle A^{−1}\) is available only when det \(\displaystyle A\ne 0\). Use the coefficient determined by the trace.
Worked solution
(a)
\(p=\operatorname{tr}(A)=21\)
Answer: \(21\).
Answer: \(21\)
(b)
\(q=\det A=109\)
Answer: \(109\).
Answer: \(109\)
(c)
\(\displaystyle A^{- 1}= \frac{pI- A}{q}= [\begin{matrix}\frac{11}{109}& - \frac{1}{109}\\ - \frac{1}{109}& \frac{10}{109}\end{matrix}]\)
Answer: \(\displaystyle - \frac{1}{109}\).
Answer: \(\displaystyle - \frac{1}{109}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Use the coefficient determined by the trace.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A scalar constant in a matrix identity is multiplied by I.
(b) · 2 marks
- 1 mark: Use the determinant as the constant term.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: An entrywise reciprocal is not a matrix inverse.
(c) · 2 marks
- 1 mark: Multiply the matrix characteristic equation by \(\displaystyle A^{−1}\).
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A scalar constant in a matrix identity is multiplied by I.
Total: 6 marks.