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Determinants — 9231 FP1

Use cofactors or elimination for a determinant and inverse.

Notes and key methods

A 3×3 determinant can be expanded along a row or found using determinant-preserving row operations. A row swap reverses its sign; multiplying a row scales it by the same factor.

An inverse exists when the determinant is nonzero. Solve \(\displaystyle AX=I\) by elimination or use cofactors and transpose the cofactor matrix. Check a proposed inverse by multiplying it on both sides.

Check the conditions

Three worked examples

Example 1 · Three-by-three matrices

\(A=\left[\begin{matrix}3 & 1 & 0\\0 & 4 & 1\\1 & 0 & 5\end{matrix}\right]\). For the system question use \(b=\left[\begin{matrix}17\\27\\39\end{matrix}\right]\).

  1. Find det A.
  2. Find the (1,1) entry of A inverse.
  3. Find the first coordinate of the solution of Av=b.
  4. Find trace(\(\displaystyle A^{2}\)).
Show the reasoning

(a)

\(\det A=3(4)(5)+1=61\)

Answer: \(61\).

(b)

\(\displaystyle A^{-1}=\left[\begin{matrix}\frac{20}{61} & - \frac{5}{61} & \frac{1}{61}\\\frac{1}{61} & \frac{15}{61} & - \frac{3}{61}\\- \frac{4}{61} & \frac{1}{61} & \frac{12}{61}\end{matrix}\right]\)

Answer: \(\displaystyle \frac{20}{61}\).

(c)

\(v=\left[\begin{matrix}4\\5\\7\end{matrix}\right]\)

Answer: \(4\).

(d)

\(\operatorname{tr}(A^2)=50\)

Answer: \(50\).

Example 2 · Three-by-three matrices

\(A=\left[\begin{matrix}4 & 1 & 0\\0 & 5 & 1\\1 & 0 & 6\end{matrix}\right]\). For the system question use \(b=\left[\begin{matrix}26\\38\\53\end{matrix}\right]\).

  1. Find det A.
  2. Find the (1,1) entry of A inverse.
  3. Find the first coordinate of the solution of Av=b.
  4. Find trace(\(\displaystyle A^{2}\)).
Show the reasoning

(a)

\(\det A=4(5)(6)+1=121\)

Answer: \(121\).

(b)

\(\displaystyle A^{-1}=\left[\begin{matrix}\frac{30}{121} & - \frac{6}{121} & \frac{1}{121}\\\frac{1}{121} & \frac{24}{121} & - \frac{4}{121}\\- \frac{5}{121} & \frac{1}{121} & \frac{20}{121}\end{matrix}\right]\)

Answer: \(\displaystyle \frac{30}{121}\).

(c)

\(v=\left[\begin{matrix}5\\6\\8\end{matrix}\right]\)

Answer: \(5\).

(d)

\(\operatorname{tr}(A^2)=77\)

Answer: \(77\).

Example 3 · Three-by-three matrices

\(A=\left[\begin{matrix}5 & 1 & 0\\0 & 6 & 1\\1 & 0 & 7\end{matrix}\right]\). For the system question use \(b=\left[\begin{matrix}37\\51\\69\end{matrix}\right]\).

  1. Find det A.
  2. Find the (1,1) entry of A inverse.
  3. Find the first coordinate of the solution of Av=b.
  4. Find trace(\(\displaystyle A^{2}\)).
Show the reasoning

(a)

\(\det A=5(6)(7)+1=211\)

Answer: \(211\).

(b)

\(\displaystyle A^{-1}=\left[\begin{matrix}\frac{42}{211} & - \frac{7}{211} & \frac{1}{211}\\\frac{1}{211} & \frac{35}{211} & - \frac{5}{211}\\- \frac{6}{211} & \frac{1}{211} & \frac{30}{211}\end{matrix}\right]\)

Answer: \(\displaystyle \frac{42}{211}\).

(c)

\(v=\left[\begin{matrix}6\\7\\9\end{matrix}\right]\)

Answer: \(6\).

(d)

\(\operatorname{tr}(A^2)=110\)

Answer: \(110\).

Common mistakes

Practise with answer checking

Original exam-style problems

Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.

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