3.3.1: Exercises 3.3

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In Exercises \(\PageIndex{1}\) - \(\PageIndex{8}\), find the determinant of the \(2\times 2\) matrix.

Exercise \(\PageIndex{1}\)

\(\left[\begin{array}{cc}{10}&{7}\\{8}&{9}\end{array}\right]\)

Answer: \(34\)

Exercise \(\PageIndex{2}\)

\(\left[\begin{array}{cc}{6}&{-1}\\{-7}&{8}\end{array}\right]\)

Answer: \(41\)

Exercise \(\PageIndex{3}\)

\(\left[\begin{array}{cc}{-1}&{-7}\\{-5}&{9}\end{array}\right]\)

Answer: \(-44\)

Exercise \(\PageIndex{4}\)

\(\left[\begin{array}{cc}{-10}&{-1}\\{-4}&{7}\end{array}\right]\)

Answer: \(-74\)

Exercise \(\PageIndex{5}\)

\(\left[\begin{array}{cc}{8}&{10}\\{2}&{-3}\end{array}\right]\)

Answer: \(-44\)

Exercise \(\PageIndex{6}\)

\(\left[\begin{array}{cc}{10}&{-10}\\{-10}&{0}\end{array}\right]\)

Answer: \(-100\)

Exercise \(\PageIndex{7}\)

\(\left[\begin{array}{cc}{1}&{-3}\\{7}&{7}\end{array}\right]\)

Answer: \(28\)

Exercise \(\PageIndex{8}\)

\(\left[\begin{array}{cc}{-4}&{-5}\\{-1}&{-4}\end{array}\right]\)

Answer: \(11\)

In Exercises \(\PageIndex{9}\) - \(\PageIndex{12}\), a matrix \(A\) is given.

Construct the submatrices used to compute the minors \(A_{1,1}\), \(A_{1,2}\), and \(A_{1,3}\).
Find the cofactors \(C_{1,1}\), \(C_{1,2}\), and \(C_{1,3}\).

Exercise \(\PageIndex{9}\)

\(\left[\begin{array}{ccc}{-7}&{-3}&{10}\\{3}&{7}&{6}\\{1}&{6}&{10}\end{array}\right]\)

Answer

The submatrices are \(\left[\begin{array}{cc}{7}&{6}\\{6}&{10}\end{array}\right]\), \(\left[\begin{array}{cc}{3}&{6}\\{1}&{10}\end{array}\right]\), and \(\left[\begin{array}{cc}{3}&{7}\\{1}&{6}\end{array}\right]\), respectively.
\(C_{1,2} = 34\), \(C_{1,2} = −24\), \(C_{1,3} = 11\)

Exercise \(\PageIndex{10}\)

\(\left[\begin{array}{ccc}{-2}&{-9}&{6}\\{-10}&{-6}&{8}\\{0}&{-3}&{-2}\end{array}\right]\)

Answer

The submatrices are \(\left[\begin{array}{cc}{-6}&{8}\\{-3}&{-2}\end{array}\right]\), \(\left[\begin{array}{cc}{-10}&{8}\\{0}&{-2}\end{array}\right]\), and \(\left[\begin{array}{cc}{10}&{-6}\\{0}&{-3}\end{array}\right]\), respectively.
\(C_{1,2} = 36\), \(C_{1,2} = −20\), \(C_{1,3} = -30\)

Exercise \(\PageIndex{11}\)

\(\left[\begin{array}{ccc}{-5}&{-3}&{3}\\{-3}&{3}&{10}\\{-9}&{3}&{9}\end{array}\right]\)

Answer

The submatrices are \(\left[\begin{array}{cc}{3}&{10}\\{3}&{9}\end{array}\right]\), \(\left[\begin{array}{cc}{-3}&{10}\\{-9}&{9}\end{array}\right]\), and \(\left[\begin{array}{cc}{-3}&{3}\\{-9}&{3}\end{array}\right]\), respectively.
\(C_{1,2} = -3\), \(C_{1,2} = −63\), \(C_{1,3} = 18\)

Exercise \(\PageIndex{12}\)

\(\left[\begin{array}{ccc}{-6}&{-4}&{6}\\{-8}&{0}&{0}\\{-10}&{8}&{-1}\end{array}\right]\)

Answer

The submatrices are \(\left[\begin{array}{ccc}{-6}&{-4}&{6}\\{-8}&{0}&{0}\\{-10}&{8}&{-1}\end{array}\right]\), \(\left[\begin{array}{cc}{-8}&{0}\\{-10}&{-1}\end{array}\right]\), and \(\left[\begin{array}{cc}{-8}&{0}\\{-10}&{8}\end{array}\right]\), respectively.
\(C_{1,2} = 0\), \(C_{1,2} = −8\), \(C_{1,3} = -64\)

In Exercises \(\PageIndex{13}\) – \(\PageIndex{24}\), find the determinant of the given matrix using cofactor expansion along the first row.

Exercise \(\PageIndex{13}\)

\(\left[\begin{array}{ccc}{3}&{2}&{3}\\{-6}&{1}&{-10}\\{-8}&{-9}&{-9}\end{array}\right]\)

Answer: \(-59\)

Exercise \(\PageIndex{14}\)

\(\left[\begin{array}{ccc}{8}&{-9}&{-2}\\{-9}&{9}&{-7}\\{5}&{-1}&{9}\end{array}\right]\)

Answer: \(250\)

Exercise \(\PageIndex{15}\)

\(\left[\begin{array}{ccc}{-4}&{3}&{-4}\\{-4}&{-5}&{3}\\{3}&{-4}&{5}\end{array}\right]\)

Answer: \(15\)

Exercise \(\PageIndex{16}\)

\(\left[\begin{array}{ccc}{1}&{-2}&{1}\\{5}&{5}&{4}\\{4}&{0}&{0}\end{array}\right]\)

Answer: \(-52\)

Exercise \(\PageIndex{17}\)

\(\left[\begin{array}{ccc}{1}&{-4}&{1}\\{0}&{3}&{0}\\{1}&{2}&{2}\end{array}\right]\)

Answer: \(3\)

Exercise \(\PageIndex{18}\)

\(\left[\begin{array}{ccc}{3}&{-1}&{0}\\{-3}&{0}&{-4}\\{0}&{-1}&{-4}\end{array}\right]\)

Answer: \(0\)

Exercise \(\PageIndex{19}\)

\(\left[\begin{array}{ccc}{-5}&{0}&{-4}\\{2}&{4}&{-1}\\{-5}&{0}&{-4}\end{array}\right]\)

Answer: \(0\)

Exercise \(\PageIndex{20}\)

\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{1}&{0}\\{-1}&{1}&{1}\end{array}\right]\)

Answer: \(1\)

Exercise \(\PageIndex{21}\)

\(\left[\begin{array}{cccc}{0}&{0}&{-1}&{-1}\\{1}&{1}&{0}&{1}\\{1}&{1}&{-1}&{0}\\{-1}&{0}&{1}&{0}\end{array}\right]\)

Answer: \(0\)

Exercise \(\PageIndex{22}\)

\(\left[\begin{array}{cccc}{-1}&{0}&{0}&{-1}\\{-1}&{0}&{0}&{1}\\{1}&{1}&{1}&{0}\\{1}&{0}&{-1}&{-1}\end{array}\right]\)

Answer: \(2\)

Exercise \(\PageIndex{23}\)

\(\left[\begin{array}{cccc}{-5}&{1}&{0}&{0}\\{-3}&{-5}&{2}&{5}\\{-2}&{4}&{-3}&{4}\\{5}&{4}&{-3}&{3}\end{array}\right]\)

Answer: \(-113\)

Exercise \(\PageIndex{24}\)

\(\left[\begin{array}{cccc}{2}&{-1}&{4}&{4}\\{3}&{-3}&{3}&{2}\\{0}&{4}&{-5}&{1}\\{-2}&{-5}&{-2}&{-5}\end{array}\right]\)

Answer: \(179\)

Exercise \(\PageIndex{25}\)

Let \(A\) be a \(2\times 2\) matrix;

\[A=\left[\begin{array}{cc}{a}&{b}\\{c}&{d}\end{array}\right]. \nonumber \]

Show why \(\text{det}(A)=ad-bc\) by computing the cofactor expansion of \(A\) along the first row.

Answer: Hint: \(C_{1,1}=d\).