3.1E: The Cofactor Expansion Exercises

Exercises for 1

solutions

2

Compute the determinants of the following matrices.

\(\left[ \begin{array}{rr} 2 & -1 \\ 3 & 2 \end{array} \right]\) \(\left[ \begin{array}{rr} 6 & 9 \\ 8 & 12 \end{array} \right]\) \(\left[ \begin{array}{rr} a^2 & ab \\ ab & b^2 \end{array} \right]\) \(\left[ \begin{array}{cc} a+1 & a \\ a & a-1 \end{array} \right]\) \(\left[ \begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right]\) \(\left[ \begin{array}{rrr} 2 & 0 & -3 \\ 1 & 2 & 5 \\ 0 & 3 & 0 \end{array} \right]\) \(\left[ \begin{array}{rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right]\) \(\left[ \begin{array}{rrr} 0 & a & 0 \\ b & c & d \\ 0 & e & 0 \end{array} \right]\) \(\left[ \begin{array}{rrr} 1 & b & c \\ b & c & 1 \\ c & 1 & b \end{array} \right]\) \(\left[ \begin{array}{rrr} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right]\) \(\left[ \begin{array}{rrrr} 0 & 1 & -1 & 0 \\ 3 & 0 & 0 & 2 \\ 0 & 1 & 2 & 1 \\ 5 & 0 & 0 & 7 \end{array} \right]\) \(\left[ \begin{array}{rrrr} 1 & 0 & 3 & 1 \\ 2 & 2 & 6 & 0 \\ -1 & 0 & -3 & 1 \\ 4 & 1 & 12 & 0 \end{array} \right]\) \(\left[ \begin{array}{rrrr} 3 & 1 & -5 & 2 \\ 1 & 3 & 0 & 1 \\ 1 & 0 & 5 & 2 \\ 1 & 1 & 2 & -1 \end{array} \right]\) \(\left[ \begin{array}{rrrr} 4 & -1 & 3 & -1 \\ 3 & 1 & 0 & 2 \\ 0 & 1 & 2 & 2 \\ 1 & 2 & -1 & 1 \end{array} \right]\) \(\left[ \begin{array}{rrrr} 1 & -1 & 5 & 5 \\ 3 & 1 & 2 & 4 \\ -1 & -3 & 8 & 0 \\ 1 & 1 & 2 & -1 \end{array} \right]\) \(\left[ \begin{array}{rrrr} 0 & 0 & 0 & a \\ 0 & 0 & b & p \\ 0 & c & q & k \\ d & s & t & u \end{array} \right]\)

2. \(0\)
3. \(-1\)
4. \(-39\)
5. \(0\)
6. \(2abc\)
7. \(0\)
8. \(-56\)
9. \(abcd\)

Show that \(\det A = 0\) if \(A\) has a row or column consisting of zeros.

Show that the sign of the position in the last row and the last column of \(A\) is always \(+1\).

Show that \(\det I = 1\) for any identity matrix \(I\).

Evaluate the determinant of each matrix by reducing it to upper triangular form.

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

2. \(-17\)
3. \(106\)

Evaluate by cursory inspection:

1. \(\det \left[ \begin{array}{ccc} a & b & c \\ a+1 & b+1 & c+1 \\ a-1 & b-1 & c-1 \end{array} \right]\)
2. \(\det \left[ \begin{array}{ccc} a & b & c \\ a+b & 2b & c+b \\ 2 & 2 & 2 \end{array} \right]\)

2. \(0\)

If \(\det \left[ \begin{array}{rrr} a & b & c \\ p & q & r \\ x & y & z \end{array} \right] = -1\) compute:

1. \(\det \left[ \begin{array}{ccc} -x & -y & -z \\ 3p+a & 3q+b & 3r+c \\ 2p & 2q & 2r \end{array} \right]\)
2. \(\det \left[ \begin{array}{ccc} -2a & -2b & -2c \\ 2p+x & 2q+y & 2r+z \\ 3x & 3y & 3z \end{array} \right]\)

2. \(12\)

Show that:

1. \(\det \left[ \begin{array}{rrr} p+x & q+y & r+z \\ a+x & b+y & c+z \\ a+p & b+q & c+r \end{array} \right] = 2\det \left[ \begin{array}{rrr} a & b & c \\ p & q & r \\ x & y & z \end{array} \right]\)
2. \(\det \left[ \begin{array}{rrr} 2a+p & 2b+q & 2c+r \\ 2p+x & 2q+y & 2r+z \\ 2x+a & 2y+b & 2z+c \end{array} \right] = 9\det \left[ \begin{array}{rrr} a & b & c \\ p & q & r \\ x & y & z \end{array} \right]\)

2. \(\det \left[ \begin{array}{rrr} 2a+p & 2b+q & 2c+r \\ 2p+x & 2q+y & 2r+z \\ 2x+a & 2y+b & 2z+c \end{array} \right]\)
   \(= 3 \det \left[ \begin{array}{rrr} a+p+x & b+q+y & c+r+z \\ 2p+x & 2q+y & 2r+z \\ 2x+a & 2y+b & 2z+c \end{array} \right]\)
   \(= 3 \det \left[ \begin{array}{rrr} a+p+x & b+q+y & c+r+z \\ p-a & q-b & r-c \\ x-p & y-q & z-r \end{array} \right]\)
   \(= 3 \det \left[ \begin{array}{rrr} 3x & 3y & 3z \\ p-a & q-b & r-c \\ x-p & y-q & z-r \end{array} \right] \cdots\)

In each case either prove the statement or give an example showing that it is false:

1. \(\det (A + B) = \det A + \det B.\)
2. If \(\det A = 0\), then \(A\) has two equal rows.
3. If \(A\) is \(2 \times 2\), then \(\det (A^T) = \det A\).
4. If \(R\) is the reduced row-echelon form of \(A\), then \(\det A = \det R\).
5. If \(A\) is \(2 \times 2\), then \(\det (7A) = 49 \det A\).
6. \(\det (A^T) = - \det A\).
7. \(\det (-A) = - \det A\).
8. If \(\det A = \det B\) where \(A\) and \(B\) are the same size, then \(A\) = \(B\).

2. False. \(A = \left[ \begin{array}{rr} 1 & 1 \\ 2 & 2 \end{array} \right]\)
3. False. \(A = \left[ \begin{array}{rr} 2 & 0 \\ 0 & 1 \end{array} \right] \rightarrow R = \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right]\)
4. False. \(A = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right]\)
5. False. \(A = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right]\) and \(B = \left[ \begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array} \right]\)

Compute the determinant of each matrix, using Theorem [thm:007890].

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

2. \(35\)

If \(\det A = 2, \det B = -1\), and \(\det C = 3\), find:

\(\det \left[ \begin{array}{ccc} A & X & Y \\ 0 & B & Z \\ 0 & 0 & C \end{array}\right]\) \(\det \left[ \begin{array}{ccc} A & 0 & 0 \\ X & B & 0 \\ Y & Z & C \end{array}\right]\) \(\det \left[ \begin{array}{ccc} A & X & Y \\ 0 & B & 0 \\ 0 & Z & C \end{array}\right]\) \(\det \left[ \begin{array}{ccc} A & X & 0 \\ 0 & B & 0 \\ Y & Z & C \end{array}\right]\)

2. \(-6\)
3. \(-6\)

If \(A\) has three columns with only the top two entries nonzero, show that \(\det A = 0\).

1. Find \(\det A\) if \(A\) is \(3 \times 3\) and \(\det (2A) = 6\).
2. Under what conditions is \(\det (-A) = \det A\)?

Evaluate by first adding all other rows to the first row.

1. \(\det \left[ \begin{array}{ccc} x-1 & 2 & 3 \\ 2 & -3 & x-2 \\ -2 & x & -2 \end{array}\right]\)
2. \(\det \left[ \begin{array}{ccc} x-1 & -3 & 1 \\ 2 & -1 & x-1 \\ -3 & x+2 & -2 \end{array}\right]\)

2. \(-(x-2)(x^2 + 2x-12)\)

1. Find \(b\) if \(\det \left[ \begin{array}{rrr} 5 & -1 & x \\ 2 & 6 & y \\ -5 & 4 & z \end{array}\right] = ax + by + cz\).
2. Find \(c\) if \(\det \left[ \begin{array}{rrr} 2 & x & -1 \\ 1 & y & 3 \\ -3 & z & 4 \end{array}\right] = ax + by + cz\).

2. \(-7\)

Find the real numbers \(x\) and \(y\) such that \(\det A = 0\) if:

\(A = \left[ \begin{array}{rrr} 0 & x & y \\ y & 0 & x \\ x & y & 0 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 1 & x & x \\ -x & -2 & x \\ -x & -x & -3 \end{array}\right]\) \(A = \left[ \begin{array}{rrrr} 1 & x & x^2 & x^3 \\ x & x^2 & x^3 & 1 \\ x^2 & x^3 & 1 & x \\ x^3 & 1 & x & x^2 \end{array}\right]\) \(A = \left[ \begin{array}{rrrr} x & y & 0 & 0 \\ 0 & x & y & 0 \\ 0 & 0 & x & y \\ y & 0 & 0 & x \end{array}\right]\)

2. \(\pm \frac{\sqrt{6}}{2}\)
3. \(x = \pm y\)

Show that
\(\det \left[ \begin{array}{rrrr} 0 & 1 & 1 & 1 \\ 1 & 0 & x & x \\ 1 & x & 0 & x \\ 1 & x & x & 0 \end{array}\right] = -3x^2\)

Show that
\(\det \left[ \begin{array}{rrrr} 1 & x & x^2 & x^3 \\ a & 1 & x & x^2 \\ p & b & 1 & x \\ q & r & c & 1 \end{array}\right] = (1-ax)(1-bx)(1-cx).\)

[ex:3.1.19] Given the polynomial \(p(x) = a + bx + cx^2 + dx^3 + x^4\), the matrix \(C = \left[ \begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -a & -b & -c & -d \end{array}\right]\) is called the companion matrix of \(p(x)\). Show that \(\det (xI - C) = p(x)\).

Show that
\(\det \left[ \begin{array}{rrr} a+x & b+x & c+x \\ b+x & c+x & a+x \\ c+x & a+x & b+x \end{array}\right] \\ = (a+b+c+3x)[(ab+ac+bc)-(a^2+b^2+c^2)]\)

[ex:3_1_21]. Prove Theorem [thm:007914]. [Hint: Expand the determinant along column j.]

Let \(\mathbf{x} = \left[ \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \right]\), \(\mathbf{y} = \left[ \begin{array}{c} y_1 \\ y_2 \\ \vdots \\ y_n \end{array} \right]\) and \(A = \left[ \begin{array}{ccccc} \mathbf{c}_1 & \cdots & \mathbf{x} + \mathbf{y}& \cdots & \mathbf{c}_n \end{array}\right]\) where \(\mathbf{x} + \mathbf{y}\) is in column \(j\). Expanding \(\det A\) along column \(j\) (the one containing \(\mathbf{x} + \mathbf{y}\)):

\[\begin{aligned} T(\mathbf{x} + \mathbf{y}) = \det A &= \sum_{i=1}^{n} (x_i + y_i)c_{ij}(A) \\ &= \sum_{i=1}^{n} x_ic_{ij}(A) + \sum_{i=1}^{n} y_ic_{ij}(A)\\ &= T(\mathbf{x}) + T(\mathbf{y})\end{aligned} \nonumber \]

Similarly for \(T(a\mathbf{x}) = aT(\mathbf{x})\).

Show that

\[\det \left[ \begin{array}{ccccc} 0 & 0 & \cdots & 0 & a_1 \\ 0 & 0 & \cdots & a_2 & * \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & a_{n-1} & \cdots & * & * \\ a_n & * & \cdots & * & * \end{array} \right] = (-1)^k a_1a_2 \cdots a_n \nonumber \]

where either \(n = 2k\) or \(n = 2k + 1\), and \(*\)-entries are arbitrary.

By expanding along the first column, show that:

\[\det \left[ \begin{array}{ccccccc} 1 & 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & 1 & 1 \\ 1 & 0 & 0 & 0 & \cdots & 0 & 1 \\ \end{array} \right] = 1 + (-1)^{n+1} \nonumber \]

if the matrix is \(n \times n, n \geq 2\).

Form matrix \(B\) from a matrix \(A\) by writing the columns of \(A\) in reverse order. Express \(\det B\) in terms of \(\det A\).

If \(A\) is \(n \times n\), then \(\det B = (-1)^k \det A\) where \(n = 2k\) or \(n = 2k + 1\).

Prove property 3 of Theorem [thm:007779] by expanding along the row (or column) in question.

Show that the line through two distinct points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) in the plane has equation

\[\det \left[ \begin{array}{ccc} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{array} \right] = 0 \nonumber \]

Let \(A\) be an \(n \times n\) matrix. Given a polynomial \(p(x) = a_0 + a_1x + \cdots + a_mx^m\), we write
\(p(A) = a_{0}I + a_1A + \cdots + a_mA^m\).

For example, if \(p(x) = 2-3x+5x^2\), then
\(p(A) = 2I -3A +5A^2\). The characteristic polynomial of \(A\) is defined to be \(c_A(x) = \det [xI - A]\), and the Cayley-Hamilton theorem asserts that \(c_A(A) = 0\) for any matrix \(A\).

1. 2
   1. \(A = \left[ \begin{array}{rr} 3 & 2 \\ 1 & -1 \end{array}\right]\)
   2. \(A = \left[ \begin{array}{rrr} 1 & -1 & 1 \\ 0 & 1 & 0 \\ 8 & 2 & 2 \end{array}\right]\)
2. Prove the theorem for \(A = \left[ \begin{array}{rr} a & b \\ c & d \end{array}\right]\)