3.2E: Exercises for Section 3.2

Exercise \(\PageIndex{1}\)

An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant. \[\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\to\cdots\to\left[\begin{array}{cc}a&c\\b&d\end{array}\right]\nonumber\]

Answer

It does not change the determinant. This was just taking the transpose.

Exercise \(\PageIndex{2}\)

An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant. \[\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\to\cdots\to\left[\begin{array}{cc}c&d\\a&b\end{array}\right]\nonumber\]

Answer

In this case two rows were switched and so the resulting determinant is \(−1\) times the first

Exercise \(\PageIndex{3}\)

An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant. \[\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\to\cdots\to\left[\begin{array}{cc}a&b\\a+c&b+d\end{array}\right]\nonumber\]

Answer

The determinant is unchanged. It was just the first row added to the second.

Exercise \(\PageIndex{4}\)

An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant. \[\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\to\cdots\to\left[\begin{array}{cc}a&b\\2c&2d\end{array}\right]\nonumber\]

Answer

The second row was multiplied by \(2\) so the determinant of the result is \(2\) times the original determinant.

Exercise \(\PageIndex{5}\)

An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant. \[\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\to\cdots\to\left[\begin{array}{cc}b&a\\d&c\end{array}\right]\nonumber\]

Answer

In this case the two columns were switched so the determinant of the second is \(−1\) times the determinant of the first.

Exercise \(\PageIndex{6}\)

Let \(A\) be an \(r\times r\) matrix and suppose there are \(r −1\) rows (columns) such that all rows (columns) are linear combinations of these \(r −1\) rows (columns). Show \(\det(A) = 0\).

Answer

If the determinant is nonzero, then it will remain nonzero with row operations applied to the matrix. However, by assumption, you can obtain a row of zeros by doing row operations. Thus the determinant must have been zero after all.

Exercise \(\PageIndex{7}\)

Show \(\det(aA) = a^n \det(A)\) for an \(n\times n\) matrix \(A\) and scalar \(a\).

Answer

\(\det(aA) = \det(aIA) = \det(aI)\det(A) = a^n \det(A)\). The matrix which has a down the main diagonal has determinant equal to \(a^n\).

Exercise \(\PageIndex{8}\)

Construct \(2\times 2\) matrices \(A\) and \(B\) to show that the \(\det A\det B = \det(AB)\).

Answer

\[\begin{array}{c}\det\left(\left[\begin{array}{cc}1&2\\3&4\end{array}\right]\left[\begin{array}{cc}-1&2\\-5&6\end{array}\right]\right)=-8 \\ \det\left[\begin{array}{cc}1&2\\3&4\end{array}\right]\det\left[\begin{array}{cc}-1&2\\-5&6\end{array}\right]=-2\times 4=-8\end{array}\nonumber\]

Exercise \(\PageIndex{9}\)

Is it true that \(\det(A+B) = \det(A)+\det(B)\)? If this is so, explain why. If it is not so, give a counter example.

Answer

This is not true at all. Consider \(A=\left[\begin{array}{cc}1&0\\0&1\end{array}\right],\: B=\left[\begin{array}{cc}-1&0\\0&-1\end{array}\right]\).

Exercise \(\PageIndex{10}\)

An \(n\times n\) matrix is called nilpotent if for some positive integer, \(k\) it follows \(A^k = 0\). If \(A\) is a nilpotent matrix and \(k\) is the smallest possible integer such that \(A^k = 0\), what are the possible values of \(\det(A)\)?

Answer

It must be \(0\) because \(0 = \det(0) = \det (A^k) = (\det(A))^k\).

Exercise \(\PageIndex{11}\)

A matrix is said to be orthogonal if \(A^TA = I\). Thus the inverse of an orthogonal matrix is just its transpose. What are the possible values of \(\det(A)\) if \(A\) is an orthogonal matrix?

Answer

You would need \(\det (AA^T) = \det(A)\det (A^T) = \det(A)^2 = 1\) and so \(\det(A) = 1\), or \(-1\).

Exercise \(\PageIndex{12}\)

Let \(A\) and \(B\) be two \(n\times n\) matrices. \(A ∼ B\) (\(A\) is similar to \(B\)) means there exists an invertible matrix \(P\) such that \(A = P^{−1}BP\). Show that if \(A ∼ B\), then \(\det(A) = \det(B)\).

Answer

\(\det(A) = \det(S^{−1}BS) = \det(S^{−1})\det(B)\det(S) = \det(B)\det(S^{−1}S) = \det(B)\).

Exercise \(\PageIndex{13}\)

Tell whether each statement is true or false. If true, provide a proof. If false, provide a counter example.

1. If A is a \(3\times 3\) matrix with a zero determinant, then one column must be a multiple of some other column.
2. If any two columns of a square matrix are equal, then the determinant of the matrix equals zero.
3. For two \(n\times n\) matrices \(A\) and \(B\), \(\det(A+B) = \det(A) +\det(B)\).
4. For an \(n\times n\) matrix \(A\), \(\det(3A) = 3 \det(A)\)
5. If \(A^{−1}\) exists then \(\det(A^{−1}) = \det(A)^{−1}\).
6. If \(B\) is obtained by multiplying a single row of \(A\) by \(4\) then \(\det(B) = 4 \det(A)\).
7. For \(A\) an \(n\times n\) matrix, \(\det(−A) = (−1)^n \det(A)\).
8. If \(A\) is a real \(n\times n\) matrix, then \(\det (A^TA) ≥ 0\).
9. If \(A^k = 0\) for some positive integer \(k\), then \(\det(A) = 0\).
10. If \(AX = 0\) for some \(X\neq 0\), then \(\det(A) = 0\).

Answer

1. False. Consider \(\left[\begin{array}{ccc}1&1&2\\-1&5&4\\0&3&3\end{array}\right]\)
2. True.
3. False.
4. False.
5. True.
6. False.
7. True.
8. True.
9. True.
10. True.

Exercise \(\PageIndex{14}\)

Find the determinant using row operations to first simplify. \[\left|\begin{array}{ccc}1&2&1\\2&3&2\\-4&1&2\end{array}\right|\nonumber\]

Answer

\[\left|\begin{array}{ccc}1&2&1\\2&3&2\\-4&1&2\end{array}\right|=-6\nonumber\]

Exercise \(\PageIndex{15}\)

Find the determinant using row operations to first simplify. \[\left|\begin{array}{ccc}2&1&3\\2&4&2\\1&4&-5\end{array}\right|\nonumber\]

Answer

\[\left|\begin{array}{ccc}2&1&3\\2&4&2\\1&4&-5\end{array}\right|=-32\nonumber\]

Exercise \(\PageIndex{16}\)

Find the determinant using row operations to first simplify. \[\left|\begin{array}{cccc}1&2&1&2\\3&1&-2&3\\-1&0&3&1\\2&3&2&-2\end{array}\right|\nonumber\]

Answer

One can row reduce this using only row operation 3 to \[\left[\begin{array}{cccc}1&2&1&2\\0&-5&-5&-3 \\ 0&0&2&\frac{9}{5} \\ 0&0&0&-\frac{63}{10}\end{array}\right]\nonumber\] and therefore, the determinant is \(-63\). \[\left|\begin{array}{cccc}1&2&1&2\\3&1&-2&3\\-1&0&3&1\\2&3&2&-2\end{array}\right|=63\nonumber\]

Exercise \(\PageIndex{17}\)

Find the determinant using row operations to first simplify. \[\left|\begin{array}{cccc}1&4&1&2\\3&2&-2&3\\-1&0&3&3\\2&1&2&-2\end{array}\right|\nonumber\]

Answer

One can row reduce this using only row operation 3 to \[\left[\begin{array}{cccc}1&4&1&2\\0&-10&-5&-3 \\ 0&0&2&\frac{19}{5} \\ 0&0&0&-\frac{211}{20}\end{array}\right]\nonumber\] Thus the determinant is given by \[\left|\begin{array}{cccc}1&4&1&2\\3&2&-2&3\\-1&0&3&3\\2&1&2&-2\end{array}\right|=211\nonumber\]