3.2E: Determinants and Matrix Inverses Exercises

Exercise \(\PageIndex{1}\)

Find the adjugate of each of the following matrices.

1. \(\left[ \begin{array}{rrr} 5 & 1 & 3 \\ -1 & 2 & 3 \\ 1 & 4 & 8 \end{array}\right]\)
2. \(\left[ \begin{array}{rrr} 1 & -1 & 2 \\ 3 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right]\)
3. \(\left[ \begin{array}{rrr} 1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right]\)
4. \(\frac{1}{3}\left[ \begin{array}{rrr} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{array}\right]\)

Answer

2. \(\left[ \begin{array}{rrr} 1 & -1 & -2 \\ -3 & 1 & 6 \\ -3 & 1 & 4 \end{array}\right]\)
4. \(\frac{1}{3}\left[ \begin{array}{rrr} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{array}\right] = A\)

Exercise \(\PageIndex{2}\)

Use determinants to find which real values of \(c\) make each of the following matrices invertible.

1. \(\left[ \begin{array}{rrr} 1 & 0 & 3 \\ 3 & -4 & c \\ 2 & 5 & 8 \end{array}\right]\)
2. \(\left[ \begin{array}{rrr} 0 & c & -c \\ -1 & 2 & 1 \\ c & -c & c \end{array}\right]\)
3. \(\left[ \begin{array}{rrr} c & 1 & 0 \\ 0 & 2 & c \\ -1 & c & 5 \end{array}\right]\)
4. \(\left[ \begin{array}{rrr} 4 & c & 3 \\ c & 2 & c \\ 5 & c & 4 \end{array}\right]\)
5. \(\left[ \begin{array}{rrr} 1 & 2 & -1 \\ 0 & -1 & c \\ 2 & c & 1 \end{array}\right]\) \(\left[ \begin{array}{rrr} 1 & c & -1 \\ c & 1 & 1 \\ 0 & 1 & c \end{array}\right]\)

Answer

1. \(c \neq 0\)
4. any \(c\)
6. \(c \neq -1\)

Exercise \(\PageIndex{3}\)

Let \(A\), \(B\), and \(C\) denote \(n \times n\) matrices and assume that \(\det A = -1\), \(\det B = 2\), and \(\det C = 3\). Evaluate:

1. \(\det (A^{3}BC^{T}B^{-1})\)
2. \(\det (B^{2}C^{-1}AB^{-1}C^{T})\)

Answer

2. \(-2\)

Exercise \(\PageIndex{4}\)

Let \(A\) and \(B\) be invertible \(n \times n\) matrices. Evaluate:

1. \(\det (B^{-1}AB)\)
2. \(\det (A^{-1}B^{-1}AB)\)

Answer

2. \(1\)

Exercise \(\PageIndex{5}\)

If \(A\) is \(3 \times 3\) and \(\det (2A^{-1}) = -4\) and \(\det (A^{3}(B^{-1})^{T})=-4\), find \(\det A\) and \(\det B\)

Exercise \(\PageIndex{6}\)

Let \(A = \left[ \begin{array}{rrr} a & b & c \\ p & q & r \\ u & v & w \end{array}\right]\) and assume that \(\det A = 3\). Compute:

1. \(\det (2B^{-1}) \mbox{ where } B = \left[ \begin{array}{rrr} 4u & 2a & -p \\ 4v & 2b & -q \\ 4w & 2c & -r \end{array}\right]\)
2. \(\det (2C^{-1}) \mbox{ where } C = \left[ \begin{array}{ccc} 2p & -a+u & 3u \\ 2q & -b+v & 3v \\ 2r & -c+w & 3w \end{array}\right]\)

Answer

2. \(\frac{4}{9}\)

Exercise \(\PageIndex{7}\)

If \(\det \left[ \begin{array}{rr} a & b \\ c& d \end{array}\right] = -2\) calculate:

1. \(\det \left[ \begin{array}{ccc} 2 & -2 & 0 \\ c+1 & -1 & 2a \\ d-2 & 2 & 2b \end{array}\right]\)
2. \(\det \left[ \begin{array}{ccc} 2b & 0 & 4d \\ 1 & 2 & -2 \\ a+1 & 2 & 2(c-1) \end{array}\right]\)
3. \(\det (3A^{-1}) \mbox{ where } A = \left[ \begin{array}{rr} 3c & a+c \\ 3d & b+d \end{array}\right]\)

Answer

2. \(16\)

Exercise \(\PageIndex{8}\)

Solve each of the following by Cramer’s rule:

1. \( \begin{array}{rrrrr} 2x & + & y & = & 1\\ 3x & + & 7y & = & -2 \end{array}\)
2. \( \begin{array}{rrrrr} 3x & + & 4y & = & 9 \\ 2x & - & y & = & -1 \end{array}\) \( \begin{array}{rrrrrrr} 5x & + & y & - & z & = & -7 \\ 2x & - & y & - & 2z & = & 6 \\ 3x & & & + & 2z & = & -7 \end{array}\)
3. \( \begin{array}{rrrrrrr} 4x & - & y & + & 3z & = & 1 \\ 6x & + & 2y & - & z & = & 0 \\ 3x & + & 3y & + & 2z & = & -1 \end{array}\)

Answer

2. \(\frac{1}{11}\left[ \begin{array}{r} 5 \\ 21 \end{array}\right]\)
4. \(\frac{1}{79}\left[ \begin{array}{r} 12 \\ -37 \\ -2 \end{array}\right]\)

Exercise \(\PageIndex{9}\)

Use Theorem 3.2.4 to find the \((2, 3)\)-entry of \(A^{-1}\) if:

1. \(A = \left[ \begin{array}{rrr} 3 & 2 & 1 \\ 1 & 1 & 2 \\ -1 & 2 & 1 \end{array}\right]\)
2. \(A = \left[ \begin{array}{rrr} 1 & 2 & -1 \\ 3 & 1 & 1 \\ 0 & 4 & 7 \end{array}\right]\)

Answer

2. \(\frac{4}{51}\)

Exercise \(\PageIndex{10}\)

Explain what can be said about \(\det A\) if:

1. \(A^{2} = A\) \(A^{2} = I\) \(A^{3} = A\) \(PA = P\) and \(P\) is invertible
2. \(A^{2} = uA\) and \(A\) is \(n \times n\) \(A = -A^{T}\) and \(A\) is \(n \times n\)
3. \(A^{2} + I = 0\) and \(A\) is \(n \times n\)

Answer

2. \(\det A = 1, -1\)
4. \(\det A = 1\)
6. \(\det A = 0\) if \(n\) is odd; nothing can be said if \(n\) is even

Exercise \(\PageIndex{11}\)

Let \(A\) be \(n \times n\). Show that \(uA = (uI)A\), and use this with Theorem 3.2.1 to deduce the result in Theorem 3.1.2: \(\det (uA) = u^{n} \det A\)

Exercise \(\PageIndex{12}\)

If \(A\) and \(B\) are \(n \times n\) matrices, if \(AB = -BA\), and if \(n\) is odd, show that either \(A\) or \(B\) has no inverse

Exercise \(\PageIndex{13}\)

Show that \(\det AB = \det BA\) holds for any two \(n \times n\) matrices \(A\) and \(B\)

Exercise \(\PageIndex{14}\)

If \(A^{k} = 0\) for some \(k \geq 1\), show that \(A\) is not invertible

Exercise \(\PageIndex{15}\)

If \(A^{-1} = A^{T}\), describe the cofactor matrix of \(A\) in terms of \(A\).

Answer

\(dA\) where \(d = \det A\)

Exercise \(\PageIndex{16}\)

Show that no \(3 \times 3\) matrix \(A\) exists such that \(A^{2} + I = 0\). Find a \(2 \times 2\) matrix \(A\) with this property.

Exercise \(\PageIndex{17}\)

Show that \(\det (A + B^{T}) = \det (A^{T} + B)\) for any \(n \times n\) matrices \(A\) and \(B\).

Exercise \(\PageIndex{18}\)

Let \(A\) and \(B\) be invertible \(n \times n\) matrices. Show that \(\det A = \det B\) if and only if \(A = UB\) where \(U\) is a matrix with \(\det U = 1\).

Exercise \(\PageIndex{19}\)

For each of the matrices in Exercise 2, find the inverse for those values of \(c\) for which it exists.

1. \(\left[ \begin{array}{rrr} 1 & 0 & 3 \\ 3 & -4 & c \\ 2 & 5 & 8 \end{array}\right]\)
2. \(\left[ \begin{array}{rrr} 0 & c & -c \\ -1 & 2 & 1 \\ c & -c & c \end{array}\right]\)
3. \(\left[ \begin{array}{rrr} c & 1 & 0 \\ 0 & 2 & c \\ -1 & c & 5 \end{array}\right]\)
4. \(\left[ \begin{array}{rrr} 4 & c & 3 \\ c & 2 & c \\ 5 & c & 4 \end{array}\right]\)
5. \(\left[ \begin{array}{rrr} 1 & 2 & -1 \\ 0 & -1 & c \\ 2 & c & 1 \end{array}\right]\) \(\left[ \begin{array}{rrr} 1 & c & -1 \\ c & 1 & 1 \\ 0 & 1 & c \end{array}\right]\)

Answer

2. \(\frac{1}{c} \left[ \begin{array}{rrr} 1 & 0 & 1 \\ 0 & c & 1 \\ -1 & c & 1 \end{array} \right], c \neq 0\)
4. \(\frac{1}{2}\left[ \begin{array}{rrr} 8-c^2 & -c & c^2-6 \\ c & 1 & -c \\ c^2-10 & c & 8-c^2 \end{array} \right]\)
6. \(\frac{1}{c^3+1} \left[ \begin{array}{rrr} 1-c & c^2+1 & -c-1 \\ c^2 & -c & c+1 \\ -c & 1 & c^2-1 \end{array} \right], c \neq -1\)

Exercise \(\PageIndex{20}\)

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

1. If \(\text{adj} A\) exists, then \(A\) is invertible.
2. If \(A\) is invertible and \(\text{adj} A = A^{-1}\), then \(\det A = 1\).
3. \(\det (AB) = \det (B^{T}A)\).
4. If \(\det A \neq 0\) and \(AB = AC\), then \(B = C\).
5. If \(A^{T} = -A\), then \(\det A = -1\).
6. If \(\text{adj} A = 0\), then \(A = 0\).
7. If \(A\) is invertible, then \(\text{adj} A\) is invertible.
8. If \(A\) has a row of zeros, so also does \(\text{adj} A\).
9. \(\det (A^{T}A) > 0\) for all square matrices \(A\).
10. \(\det (I + A) = 1 + \det A\).
11. If \(AB\) is invertible, then \(A\) and \(B\) are invertible.
12. If \(\det A = 1\), then \(\text{adj} A = A\).
13. If \(A\) is invertible and \(\det A = d\), then \(\text{adj} A = d A^{-1}\).

Answer

2. T. \(\det AB = \det A \det B = \det B \det A = \det BA\).
4. T. \(\det A \neq 0\) means \(A^{-1}\) exists, so \(AB = AC\) implies that \(B=C\).
6. F. If \(A = \left[ \begin{array}{rrr} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right]\) then \(\text{adj} A = 0\).
8. F. If \(A = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 0 \end{array}\right]\) then \(\text{adj} A = \left[ \begin{array}{rr} 0 & -1 \\ 0 & 1 \end{array}\right]\)
10. F. If \(A = \left[ \begin{array}{rr} -1 & 1 \\ 1 & -1 \end{array}\right]\) then \(\det (I + A) = -1\) but \(1 + \det A = 1\).
12. F. If \(A = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right]\) then \(\det A = 1\) but \(\text{adj} A = \left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array}\right] \neq A\)

Exercise \(\PageIndex{21}\)

If \(A\) is \(2 \times 2\) and \(\det A = 0\), show that one column of \(A\) is a scalar multiple of the other. [Hint: Definition 2.2.1 and Part (2) of Theorem 2.4.5.]

Exercise \(\PageIndex{22}\)

Find a polynomial \(p(x)\) of degree \(2\) such that:

1. \(p(0) = 2\), \(p(1) = 3\), \(p(3) = 8\)
2. \(p(0) = 5\), \(p(1) = 3\), \(p(2) = 5\)

Answer

2. \(5 - 4x + 2x^{2}\).

Exercise \(\PageIndex{23}\)

Find a polynomial \(p(x)\) of degree \(3\) such that:

1. \(p(0) = p(1) = 1\), \(p(-1) = 4\), \(p(2) = -5\)
2. \(p(0) = p(1) = 1\), \(p(-1) = 2\), \(p(-2) = -3\)

Answer

2. \(1- \frac{5}{3}x + \frac{1}{2} x ^2 + \frac{7}{6}x^3\)

Exercise \(\PageIndex{24}\)

Given the following data pairs, find the interpolating polynomial of degree at most \(3\) and estimate the value of \(y\) corresponding to \(x = 1.5\).

1. \((0, 1)\), \((1, 2)\), \((2, 5)\), \((3, 10)\)
2. \((0, 1)\), \((1, 1.49)\), \((2, -0.42)\), \((3, -11.33)\)
3. \((0, 2)\), \((1, 2.03)\), \((2, -0.40)\), \((-1, 0.89)\)

Answer

2. \(1 - 0.51 x + 2.1 x^2 - 1.1 x^3; 1.25\), so \(y = 1.25\)

Exercise \(\PageIndex{25}\)

If \(A = \left[ \begin{array}{rrr} 1 & a & b \\ -a & 1 & c \\ -b & -c & 1 \end{array} \right]\) show that \(\det A = 1 + a^{2} + b^{2} + c^{2}\). Hence, find \(A^{-1}\) for any \(a\), \(b\), and \(c\).

Exercise \(\PageIndex{16}\)

2. Show that \(A = \left[ \begin{array}{rrr} a & p & q \\ 0 & b & r \\ 0 & 0 & c \end{array} \right]\) has an inverse if and only if \(abc \neq 0\), and find \(A^{-1}\) in that case.
3. Show that if an upper triangular matrix is invertible, the inverse is also upper triangular.

Answer

2. Use induction on \(n\) where \(A\) is \(n \times n\). It is clear if \(n = 1\). If \(n > 1\), write \(A = \left[ \begin{array}{cc} a & X \\ 0 & B \end{array}\right]\) in block form where \(B\) is \((n - 1) \times (n - 1)\). Then \(A^{-1} = \left[ \begin{array}{cc} a^{-1} & -a^{-1}XB^{-1} \\ 0 & B^{-1} \end{array}\right]\), and this is upper triangular because \(B\) is upper triangular by induction.

Exercise \(\PageIndex{27}\)

Let \(A\) be a matrix each of whose entries are integers. Show that each of the following conditions implies the other.

1. \(A\) is invertible and \(A^{-1}\) has integer entries.
2. \(\det A = 1\) or \(-1\).

Exercise \(\PageIndex{28}\)

If \(A^{-1} = \left[ \begin{array}{rrr} 3 & 0 & 1 \\ 0 & 2 & 3 \\ 3 & 1 & -1 \end{array}\right]\) find \(\text{adj} A\).

Answer

\(-\frac{1}{21}\left[\begin{array}{rrr} 3 & 0 & 1 \\ 0 & 2 & 3\\ 3 & 1 & -1 \end{array}\right]\)

Exercise \(\PageIndex{29}\)

If \(A\) is \(3 \times 3\) and \(\det A = 2\), find \(\det (A^{-1} + 4 \text{adj} A)\).

Exercise \(\PageIndex{30}\)

Show that \(\det \left[ \begin{array}{rr} 0 & A \\ B & X \end{array}\right] = \det A \det B\) when \(A\) and \(B\) are \(2 \times 2\). What if \(A\) and \(B\) are \(3 \times 3\)?

[Hint: Block multiply by \(\left[ \begin{array}{rr} 0 & I \\ I & 0 \end{array}\right]\).]

Exercise \(\PageIndex{31}\)

Let \(A\) be \(n \times n\), \(n \geq 2\), and assume one column of \(A\) consists of zeros. Find the possible values of \(rank \;(\text{adj} A)\).

Exercise \(\PageIndex{32}\)

If \(A\) is \(3 \times 3\) and invertible, compute \(\det (-A^{2}(\text{adj} A)^{-1})\).

Exercise \(\PageIndex{33}\)

Show that \(\text{adj}(uA) = u^{n-1} \text{adj} A\) for all \(n \times n\) matrices \(A\).

Exercise \(\PageIndex{34}\)

Let \(A\) and \(B\) denote invertible \(n \times n\) matrices. Show that:

1. \(\text{adj}(\text{adj} A) = (\det A)^{n-2}A\) (here \(n \geq 2\)) [Hint: See Example 3.2.8.]
2. \(\text{adj}(A^{-1}) = (\text{adj} A)^{-1}\)
3. \(\text{adj}(A^{T}) = (\text{adj} A)^T\)
4. \(\text{adj}(AB) = (\text{adj} B)(\text{adj} A)\) [Hint: Show that \(AB \text{adj}(AB) = AB \text{adj} B \text{adj} A\).]

Answer

2. Have \((\text{adj} A)A = (\det A)I\); so taking inverses, \(A^{-1} \cdot (\text{adj} A)^{-1} = \frac{1}{\det A}I\). On the other hand, \(A^{-1} \text{adj} (A^{-1}) = \det (A^{-1})I = \frac{1}{\det A}I\). Comparison yields \(A^{-1}(\text{adj} A)^{-1} = A^{-1}\text{adj}(A^{-1})\), and part (b) follows.
4. Write \(\det A = d\), \(\det B = e\). By the adjugate formula \(AB \text{adj}(AB) = deI\), and \(AB \text{adj} B \text{adj} A = A[eI] \text{adj} A = (eI)(dI) = deI\). Done as \(AB\) is invertible.