Hint
Record the effect of each row operation on the determinant. Expand the determinant with signed minors.
Worked solution
(a)
\(\det A=9(10)(11)+1=991\)
Answer: \(991\).
Answer: \(991\)
(b)
\(\displaystyle A^{-1}=\left[\begin{matrix}\frac{110}{991} & - \frac{11}{991} & \frac{1}{991}\\\frac{1}{991} & \frac{99}{991} & - \frac{9}{991}\\- \frac{10}{991} & \frac{1}{991} & \frac{90}{991}\end{matrix}\right]\)
Answer: \(\displaystyle \frac{110}{991}\).
Answer: \(\displaystyle \frac{110}{991}\)
(c)
\(v=\left[\begin{matrix}10\\11\\13\end{matrix}\right]\)
Answer: \(10\).
Answer: \(10\)
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: Expand the determinant with signed minors.
- 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 matrix inverse is not formed by entrywise reciprocals.
(b) · 2 marks
- 1 mark: Divide the appropriate transposed cofactor by the determinant.
- 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 row operation must also be applied to the augmented right side.
(c) · 2 marks
- 1 mark: Solve the three equations or apply the verified inverse.
- 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 matrix inverse is not formed by entrywise reciprocals.
Total: 6 marks.