Hint
Record the effect of each row operation on the determinant. Expand the determinant with signed minors.
Worked solution
(a)
\(\det A=10(11)(12)+1=1321\)
Answer: \(1321\).
Answer: \(1321\)
(b)
\(\displaystyle A^{-1}=\left[\begin{matrix}\frac{132}{1321} & - \frac{12}{1321} & \frac{1}{1321}\\\frac{1}{1321} & \frac{120}{1321} & - \frac{10}{1321}\\- \frac{11}{1321} & \frac{1}{1321} & \frac{110}{1321}\end{matrix}\right]\)
Answer: \(\displaystyle \frac{132}{1321}\).
Answer: \(\displaystyle \frac{132}{1321}\)
(c)
\(v=\left[\begin{matrix}11\\12\\14\end{matrix}\right]\)
Answer: \(11\).
Answer: \(11\)
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.