Exam-Style Problems

Determinants — problem 1 · Maths4U Original · 6 marks

16989

Info

Maths4U Original · M4U-EXT-9231-FP1-T408-P01

\(A=\left[\begin{matrix}7 & 1 & 0\\0 & 8 & 1\\1 & 0 & 9\end{matrix}\right]\). For the system question use \(b=\left[\begin{matrix}65\\83\\107\end{matrix}\right]\).

Find det A. Explain your reasoning. [2]
Find the (1,1) entry of A inverse. Explain your reasoning. [2]
Find the first coordinate of the solution of Av=b. Explain your reasoning. [2]

Solution

Hint

Record the effect of each row operation on the determinant. Expand the determinant with signed minors.

Worked solution

(a)

\(\det A=7(8)(9)+1=505\)

Answer: \(505\).

Answer: \(505\)

(b)

\(\displaystyle A^{-1}=\left[\begin{matrix}\frac{72}{505} & - \frac{9}{505} & \frac{1}{505}\\\frac{1}{505} & \frac{63}{505} & - \frac{7}{505}\\- \frac{8}{505} & \frac{1}{505} & \frac{56}{505}\end{matrix}\right]\)

Answer: \(\displaystyle \frac{72}{505}\).

Answer: \(\displaystyle \frac{72}{505}\)

(c)

\(v=\left[\begin{matrix}8\\9\\11\end{matrix}\right]\)

Answer: \(8\).

Answer: \(8\)

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.

Determinants — problem 2 · Maths4U Original · 6 marks

16990

Notes & worked examples

Info

Maths4U Original · M4U-EXT-9231-FP1-T408-P02

\(A=\left[\begin{matrix}8 & 1 & 0\\0 & 9 & 1\\1 & 0 & 10\end{matrix}\right]\). For the system question use \(b=\left[\begin{matrix}82\\102\\129\end{matrix}\right]\).

Find det A. Explain your reasoning. [2]
Find the (1,1) entry of A inverse. Explain your reasoning. [2]
Find the first coordinate of the solution of Av=b. Explain your reasoning. [2]

Solution

Hint

Record the effect of each row operation on the determinant. Expand the determinant with signed minors.

Worked solution

(a)

\(\det A=8(9)(10)+1=721\)

Answer: \(721\).

Answer: \(721\)

(b)

\(\displaystyle A^{-1}=\left[\begin{matrix}\frac{90}{721} & - \frac{10}{721} & \frac{1}{721}\\\frac{1}{721} & \frac{80}{721} & - \frac{8}{721}\\- \frac{9}{721} & \frac{1}{721} & \frac{72}{721}\end{matrix}\right]\)

Answer: \(\displaystyle \frac{90}{721}\).

Answer: \(\displaystyle \frac{90}{721}\)

(c)

\(v=\left[\begin{matrix}9\\10\\12\end{matrix}\right]\)

Answer: \(9\).

Answer: \(9\)

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.