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 $A = \left[ \begin{array}{rr} a & b \\ c & d \end{array}\right]$