2.5E: Matrix Inverses Exercises
- Page ID
- 132804
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solutions
2
In each case, show that the matrices are inverses of each other.
- \(\left[ \begin{array}{rr} 3 & 5 \\ 1 & 2 \end{array} \right]\), \(\left[ \begin{array}{rr} 2 & -5 \\ -1 & 3 \end{array} \right]\)
- \(\left[ \begin{array}{rr} 3 & 0 \\ 1 & -4 \end{array} \right]\), \(\frac{1}{2}\left[ \begin{array}{rr} 4 & 0 \\ 1 & -3 \end{array} \right]\)
- \(\left[ \begin{array}{rrr} 1 & 2 & 0 \\ 0 & 2 & 3 \\ 1 & 3 & 1 \end{array} \right]\), \(\left[ \begin{array}{rrr} 7 & 2 & -6 \\ -3 & -1 & 3 \\ 2 & 1 & -2 \end{array} \right]\)
- \(\left[ \begin{array}{rr} 3 & 0 \\ 0 & 5 \end{array} \right]\), \(\left[ \begin{array}{rr} \frac{1}{3} & 0 \\ 0 & \frac{1}{5} \end{array} \right]\)
Find the inverse of each of the following matrices.
\(\left[ \begin{array}{rr} 1 & -1 \\ -1 & 3 \end{array} \right]\) \(\left[ \begin{array}{rr} 4 & 1 \\ 3 & 2 \end{array} \right]\) \(\left[ \begin{array}{rrr} 1 & 0 & -1 \\ 3 & 2 & 0 \\ -1 & -1 & 0 \end{array} \right]\) \(\left[ \begin{array}{rrr} 1 & -1 & 2 \\ -5 & 7 & -11 \\ -2 & 3 & -5 \end{array} \right]\) \(\left[ \begin{array}{rrr} 3 & 5 & 0 \\ 3 & 7 & 1 \\ 1 & 2 & 1 \end{array} \right]\) \(\left[ \begin{array}{rrr} 3 & 1 & -1 \\ 2 & 1 & 0 \\ 1 & 5 & -1 \end{array} \right]\) \(\left[ \begin{array}{rrr} 2 & 4 & 1 \\ 3 & 3 & 2 \\ 4 & 1 & 4 \end{array} \right]\) \(\left[ \begin{array}{rrr} 3 & 1 & -1 \\ 5 & 2 & 0 \\ 1 & 1 & -1 \end{array} \right]\) \(\left[ \begin{array}{rrr} 3 & 1 & 2 \\ 1 & -1 & 3 \\ 1 & 2 & 4 \end{array} \right]\) \(\left[ \begin{array}{rrrr} -1 & 4 & 5 & 2 \\ 0 & 0 & 0 & -1 \\ 1 & -2 & -2 & 0 \\ 0 & -1 & -1 & 0 \end{array} \right]\) \(\left[ \begin{array}{rrrr} 1 & 0 & 7 & 5 \\ 0 & 1 & 3 & 6 \\ 1 & -1 & 5 & 2 \\ 1 & -1 & 5 & 1 \end{array} \right]\) \(\left[ \begin{array}{rrrrr} 1 & 2 & 0 & 0 & 0 \\ 0 & 1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 1 & 7 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right]\)
- \(\frac{1}{5} \left[ \begin{array}{rr} 2 & -1 \\ -3 & 4 \end{array} \right]\)
- \(\left[ \begin{array}{rrr} 2 & -1 & 3 \\ 3 & 1 & -1 \\ 1 & 1 & -2 \end{array} \right]\)
- \(\frac{1}{10} \left[ \begin{array}{rrr} 1 & 4 & -1 \\ -2 & 2 & 2 \\ -9 & 14 & -1 \end{array} \right]\)
- \(\frac{1}{4} \left[ \begin{array}{rrr} 2 & 0 & -2 \\ -5 & 2 & 5 \\ -3 & 2 & -1 \end{array} \right]\)
- \(\left[ \begin{array}{rrrr} 0 & 0 & 1 & -2 \\ -1 & -2 & -1 & -3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & 0 \end{array} \right]\)
- \(\left[ \begin{array}{rrrrr} 1 & -2 & 6 & -30 & 210 \\ 0 & 1 & -3 & 15 & -105 \\ 0 & 0 & 1 & -5 & 35 \\ 0 & 0 & 0 & 1 & -7 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right]\)
In each case, solve the systems of equations by finding the inverse of the coefficient matrix.
\( \begin{array}[t]{rrrrr} 3x & - & y & = & 5 \\ 2x & + & 2y & = & 1 \end{array}\) \( \begin{array}[t]{rrrrr} 2x & - & 3y & = & 0 \\ x & - & 4y & = & 1 \end{array}\) \( \begin{array}[t]{rrrrrrr} x & + & y & + & 2z & = & 5 \\ x & + & y & + & z & = & 0 \\ x & + & 2y & + & 4z & = & -2 \end{array}\) \( \begin{array}[t]{rrrrrrr} x & + & 4y & + & 2z & = & 1 \\ 2x & + & 3y & + & 3z & = & -1 \\ 4x & + & y & + & 4z & = & 0 \end{array}\)
- \(\left[ \begin{array}{c} x \\ y \end{array} \right] = \frac{1}{5} \left[ \begin{array}{rr} 4 & -3 \\ 1 & -2 \end{array} \right] \left[ \begin{array}{r} 0 \\ 1 \end{array} \right] = \frac{1}{5} \left[ \begin{array}{r} -3 \\ -2 \end{array} \right]\)
- \(\left[ \begin{array}{c} x \\ y \\ z \end{array} \right] = \frac{1}{5} \left[ \begin{array}{rrr} 9 & -14 & 6 \\ 4 & -4 & 1 \\ -10 & 15 & -5 \end{array} \right] \left[ \begin{array}{r} 1 \\ -1 \\ 0 \end{array} \right] = \frac{1}{5} \left[ \begin{array}{r} 23 \\ 8 \\ -25 \end{array} \right]\)
Given \(A^{-1} = \left[ \begin{array}{rrr} 1 & -1 & 3 \\ 2 & 0 & 5 \\ -1 & 1 & 0 \end{array} \right]\):
- Solve the system of equations \(A\mathbf{x} = \left[ \begin{array}{r} 1 \\ -1 \\ 3 \end{array} \right]\).
-
Find a matrix \(B\) such that
\(AB = \left[ \begin{array}{rrr} 1 & -1 & 2 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{array} \right]\). -
Find a matrix \(C\) such that
\(CA = \left[ \begin{array}{rrr} 1 & 2 & -1 \\ 3 & 1 & 1 \end{array} \right]\).
- \(B = A^{-1}AB = \left[ \begin{array}{rrr} 4 & -2 & 1 \\ 7 & -2 & 4 \\ -1 & 2 & -1 \end{array} \right]\)
Find \(A\) when
\((3A)^{-1} = \left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array} \right]\) \((2A)^{T} = \left[ \begin{array}{rr} 1 & -1 \\ 2 & 3 \end{array} \right]^{-1}\) \((I + 3A)^{-1} = \left[ \begin{array}{rr} 2 & 0 \\ 1 & -1 \end{array} \right]\) \((I - 2A^{T})^{-1} = \left[ \begin{array}{rr} 2 & 1 \\ 1 & 1 \end{array} \right]\) \(\left(A \left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array} \right] \right)^{-1} = \left[ \begin{array}{rr} 2 & 3 \\ 1 & 1 \end{array} \right]\) \(\left(\left[ \begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array} \right] A\right)^{-1} = \left[ \begin{array}{rr} 1 & 0 \\ 2 & 2 \end{array} \right]\) \(\left(A^{T} -2I \right)^{-1} = 2 \left[ \begin{array}{rr} 1 & 1 \\ 2 & 3 \end{array} \right]\) \(\left(A^{-1} -2I \right)^{T} = -2 \left[ \begin{array}{rr} 1 & 1 \\ 1 & 0 \end{array} \right]\)
- \(\frac{1}{10} \left[ \begin{array}{rr} 3 & -2 \\ 1 & 1 \end{array} \right]\)
- \(\frac{1}{2} \left[ \begin{array}{rr} 0 & 1 \\ 1 & -1 \end{array} \right]\)
- \(\frac{1}{2} \left[ \begin{array}{rr} 2 & 0 \\ -6 & 1 \end{array} \right]\)
- \(-\frac{1}{2} \left[ \begin{array}{rr} 1 & 1 \\ 1 & 0 \end{array} \right]\)
Find \(A\) when:
\(A^{-1} = \left[ \begin{array}{rrr} 1 & -1 & 3 \\ 2 & 1 & 1 \\ 0 & 2 & -2 \end{array} \right]\) \(A^{-1} = \left[ \begin{array}{rrr} 0 & 1 & -1 \\ 1 & 2 & 1 \\ 1 & 0 & 1 \end{array} \right]\)
- \(A = \frac{1}{2} \left[ \begin{array}{rrr} 2 & -1 & 3 \\ 0 & 1 & -1 \\ -2 & 1 & -1 \end{array} \right]\)
Given \(\left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \right] = \left[ \begin{array}{rrr} 3 & -1 & 2 \\ 1 & 0 & 4 \\ 2 & 1 & 0 \end{array} \right] \left[ \begin{array}{c} y_{1} \\ y_{2} \\ y_{3} \end{array} \right]\) and \(\left[ \begin{array}{c} z_{1} \\ z_{2} \\ z_{3} \end{array} \right] = \left[ \begin{array}{rrr} 1 & -1 & 1 \\ 2 & -3 & 0 \\ -1 & 1 & -2 \end{array} \right] \left[ \begin{array}{c} y_{1} \\ y_{2} \\ y_{3} \end{array} \right]\), express the variables \(x_{1}\), \(x_{2}\), and \(x_{3}\) in terms of \(z_{1}\), \(z_{2}\), and \(z_{3}\).
- In the system \( \begin{array}{rrrrr} 3x & + & 4y & = & 7 \\ 4x & + & 5y & = & 1 \end{array}\), substitute the new variables \(x^\prime\) and \(y^\prime\) given by \( \begin{array}{rrrrr} x & = & -5x^\prime & + & 4y^\prime \\ y & = & 4x^\prime & - & 3y^\prime \end{array}\). Then find \(x\) and \(y\).
- Explain part (a) by writing the equations as \(A \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{r} 7 \\ 1 \end{array} \right]\) and \(\left[ \begin{array}{r} x \\ y \end{array} \right] = B \left[ \begin{array}{c} x^\prime \\ y^\prime \end{array} \right]\). What is the relationship between \(A\) and \(B\)?
- \(A\) and \(B\) are inverses.
In each case either prove the assertion or give an example showing that it is false.
- If \(A \neq 0\) is a square matrix, then \(A\) is invertible.
- If \(A\) and \(B\) are both invertible, then \(A + B\) is invertible.
- If \(A\) and \(B\) are both invertible, then \((A^{-1}B)^{T}\) is invertible.
- If \(A^{4} = 3I\), then \(A\) is invertible.
- If \(A^{2} = A\) and \(A \neq 0\), then \(A\) is invertible.
- If \(AB = B\) for some \(B \neq 0\), then \(A\) is invertible.
- If \(A\) is invertible and skew symmetric (\(A^{T} = -A\)), the same is true of \(A^{-1}\).
- If \(A^{2}\) is invertible, then \(A\) is invertible.
- If \(AB = I\), then \(A\) and \(B\) commute.
- False. \(\left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right] + \left[ \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right]\)
- True. \(A^{-1} = \frac{1}{3}A^{3}\)
- False. \(A = B = \left[ \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right]\)
- True. If \((A^{2})B = I\), then \(A(AB) = I\); use Theorem [thm:004553].
[ex:ex2_4_10]
- If \(A\), \(B\), and \(C\) are square matrices and \(AB = I\), \(I = CA\), show that \(A\) is invertible and \(B = C = A^{-1}\).
- If \(C^{-1} = A\), find the inverse of \(C^{T}\) in terms of \(A\).
- \((C^{T})^{-1} = (C^{-1})^{T} = A^{T}\) because \(C^{-1} = (A^{-1})^{-1} = A\).
Suppose \(CA = I_{m}\), where \(C\) is \(m \times n\) and \(A\) is \(n \times m\). Consider the system \(A\mathbf{x} = \mathbf{b}\) of \(n\) equations in \(m\) variables.
- Show that this system has a unique solution \(CB\) if it is consistent.
- If \(C = \left[ \begin{array}{rrr} 0 & -5 & 1 \\ 3 & 0 & -1 \end{array} \right]\) and \(A = \left[ \begin{array}{rr} 2 & -3 \\ 1 & -2 \\ 6 & -10 \end{array} \right]\), find \(\mathbf{x}\) (if it exists) when (i) \(\mathbf{b} = \left[ \begin{array}{r} 1 \\ 0 \\ 3 \end{array} \right]\); and (ii) \(\mathbf{b} = \left[ \begin{array}{r} 7 \\ 4 \\ 22 \end{array} \right]\).
- (ii) \(\left[ \begin{array}{c} x_{1} \\ x_{2} \end{array} \right] = \left[ \begin{array}{r} 2 \\ -1 \end{array} \right]\)
Verify that \(A = \left[ \begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array} \right]\) satisfies \(A^{2} - 3A + 2I = 0\), and use this fact to show that
\(A^{-1} = \frac{1}{2}(3I - A)\).
Let \(Q = \left[ \begin{array}{rrrr} a & -b & -c & -d \\ b & a & -d & c \\ c & d & a & -b \\ d & -c & b & a \end{array} \right]\). Compute \(QQ^{T}\) and so find \(Q^{-1}\) if \(Q \neq 0\).
Let \(U = \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]\). Show that each of \(U\), \(-U\), and \(-I_{2}\) is its own inverse and that the product of any two of these is the third.
Consider \(A = \left[ \begin{array}{rr} 1 & 1 \\ -1 & 0 \end{array} \right]\),
\(B = \left[ \begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array} \right]\), \(C = \left[ \begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 5 & 0 & 0 \end{array} \right]\). Find the inverses by computing (a) \(A^{6}\); (b) \(B^{4}\); and (c) \(C^{3}\).
- \(B^{4} = I\), so \(B^{-1} = B^{3} = \left[ \begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array} \right]\)
Find the inverse of \(\left[ \begin{array}{rrr} 1 & 0 & 1 \\ c & 1 & c \\ 3 & c & 2 \end{array} \right]\) in terms of \(c\).
\(\left[ \begin{array}{crr} c^{2} - 2 & -c & 1 \\ -c & 1 & 0 \\ 3 - c^{2} & c & -1 \end{array} \right]\)
If \(c \neq 0\), find the inverse of \(\left[ \begin{array}{rrr} 1 & -1 & 1 \\ 2 & -1 & 2 \\ 0 & 2 & c \end{array} \right]\) in terms of \(c\).
Show that \(A\) has no inverse when:
- \(A\) has a row of zeros.
- \(A\) has a column of zeros.
- each row of \(A\) sums to \(0\).
- [Hint: Corollary [cor:004537], Theorem [thm:004442].]
- If column \(j\) of \(A\) is zero, \(A\mathbf{y} = \mathbf{0}\) where \(\mathbf{y}\) is column \(j\) of the identity matrix. Use Theorem [thm:004553].
- If each column of \(A\) sums to \(0\), \(XA = 0\) where \(X\) is the row of \(1\)s. Hence \(A^{T}X^{T} = 0\) so \(A\) has no inverse by Theorem [thm:004553] (\(X^{T} \neq 0\)).
Let \(A\) denote a square matrix.
- Let \(YA = 0\) for some matrix \(Y \neq 0\). Show that \(A\) has no inverse. [Hint: Corollary [cor:004537], Theorem [thm:004442].]
- Use part (a) to show that (i) \(\left[ \begin{array}{rrr} 1 & -1 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{array} \right]\); and (ii) \(\left[ \begin{array}{rrr} 2 & 1 & -1 \\ 1 & 1 & 0 \\ 1 & 0 & -1 \end{array} \right]\) have no inverse.
[Hint: For part (ii) compare row 3 with the difference between row 1 and row 2.]
- (ii) \((-1, 1, 1)A = 0\)
If \(A\) is invertible, show that
\(A^{2} \neq 0\). \(A^{k} \neq 0\) for all
\(k = 1, 2, \dots\).
- Each power \(A^{k}\) is invertible by Theorem [thm:004442] (because \(A\) is invertible). Hence \(A^{k}\) cannot be \(0\).
Suppose \(AB = 0\), where \(A\) and \(B\) are square matrices. Show that:
- If one of \(A\) and \(B\) has an inverse, the other is zero.
- It is impossible for both \(A\) and \(B\) to have inverses.
- \((BA)^{2} = 0\).
- By (a), if one has an inverse the other is zero and so has no inverse.
Find the inverse of the \(x\)-expansion in Example [exa:003128] and describe it geometrically.
If \(A = \left[ \begin{array}{rr} a & 0 \\ 0 & 1 \end{array} \right]\), \(a > 1\), then \(A^{-1} = \left[ \begin{array}{rr} \frac{1}{a} & 0 \\ 0 & 1 \end{array} \right]\) is an x-compression because \(\frac{1}{a} < 1\).
Find the inverse of the shear transformation in Example [exa:003136] and describe it geometrically.
In each case assume that \(A\) is a square matrix that satisfies the given condition. Show that \(A\) is invertible and find a formula for \(A^{-1}\) in terms of \(A\).
- \(A^{3} - 3A + 2I = 0\).
- \(A^{4} + 2A^{3} - A - 4I = 0\).
- \(A^{-1} = \frac{1}{4} (A^3 +2A^2-1)\)
Let \(A\) and \(B\) denote \(n \times n\) matrices.
- If \(A\) and \(AB\) are invertible, show that \(B\) is invertible using only (2) and (3) of Theorem [thm:004442].
- If \(AB\) is invertible, show that both \(A\) and \(B\) are invertible using Theorem [thm:004553].
- If \(B\mathbf{x} = \mathbf{0}\), then \((AB)\mathbf{x} = (A)B\mathbf{x} = \mathbf{0}\), so \(\mathbf{x} = \mathbf{0}\) because \(AB\) is invertible. Hence \(B\) is invertible by Theorem [thm:004553]. But then \(A = (AB)B^{-1}\) is invertible by Theorem [thm:004442].
In each case find the inverse of the matrix \(A\) using Example [exa:004627].
\(A = \left[ \begin{array}{rrr} -1 & 1 & 2 \\ 0 & 2 & -1 \\ 0 & 1 & -1 \end{array} \right]\) \(A = \left[ \begin{array}{rrr} 3 & 1 & 0 \\ 5 & 2 & 0 \\ 1 & 3 & -1 \end{array} \right]\) \(A = \left[ \begin{array}{rrrr} 3 & 4 & 0 & 0 \\ 2 & 3 & 0 & 0 \\ 1 & -1 & 1 & 3 \\ 3 & 1 & 1 & 4 \end{array} \right]\) \(A = \left[ \begin{array}{rrrr} 2 & 1 & 5 & 2 \\ 1 & 1 & -1 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 1 & -2 \end{array} \right]\)
- \(\left[ \begin{array}{rr|r} 2 & -1 & 0 \\ -5 & 3 & 0 \\ \hline -13 & 8 & -1 \end{array} \right]\)
- \(\left[ \begin{array}{rr|rr} 1 & -1 & -14 & 8 \\ -1 & 2 & 16 & -9 \\ \hline 0 & 0 & 2 & -1 \\ 0 & 0 & 1 & -1 \end{array} \right]\)
If \(A\) and \(B\) are invertible symmetric matrices such that \(AB = BA\), show that \(A^{-1}\), \(AB\), \(AB^{-1}\), and \(A^{-1}B^{-1}\) are also invertible and symmetric.
Let \(A\) be an \(n \times n\) matrix and let \(I\) be the \(n \times n\) identity matrix.
- If \(A^{2} = 0\), verify that \((I - A)^{-1} = I + A\).
- If \(A^{3} = 0\), verify that \((I - A)^{-1} = I + A + A^{2}\).
- Find the inverse of \(\left[ \begin{array}{rrr} 1 & 2 & -1 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{array} \right]\).
- If \(A^{n} = 0\), find the formula for \((I - A)^{-1}\).
- If \(A^{n} = 0\), \((I - A)^{-1} = I + A + \cdots + A^{n-1}\).
[ex:ex2_4_29] Prove property 6 of Theorem [thm:004442]: If \(A\) is invertible and \(a \neq 0\), then \(aA\) is invertible and \((aA)^{-1} = \frac{1}{a}A^{-1}\)
Let \(A\), \(B\), and \(C\) denote \(n \times n\) matrices. Using only Theorem [thm:004442], show that:
- If \(A\), \(C\), and \(ABC\) are all invertible, \(B\) is invertible.
- If \(AB\) and \(BA\) are both invertible, \(A\) and \(B\) are both invertible.
- \(A[B(AB)^{-1}] = I = [(BA)^{-1}B]A\), so \(A\) is invertible by Exercise [ex:ex2_4_10].
Let \(A\) and \(B\) denote invertible \(n \times n\) matrices.
- If \(A^{-1} = B^{-1}\), does it mean that \(A = B\)? Explain.
- Show that \(A = B\) if and only if \(A^{-1}B = I\).
Let \(A\), \(B\), and \(C\) be \(n \times n\) matrices, with \(A\) and \(B\) invertible. Show that
- If \(A\) commutes with \(C\), then \(A^{-1}\) commutes with \(C\).
- If \(A\) commutes with \(B\), then \(A^{-1}\) commutes with \(B^{-1}\).
- Have \(AC = CA\). Left-multiply by \(A^{-1}\) to get \(C = A^{-1}CA\). Then right-multiply by \(A^{-1}\) to get \(CA^{-1} = A^{-1}C\).
Let \(A\) and \(B\) be square matrices of the same size.
- Show that \((AB)^{2} = A^{2}B^{2}\) if \(AB = BA\).
- If \(A\) and \(B\) are invertible and \((AB)^{2} = A^{2}B^{2}\), show that \(AB = BA\).
- If \(A = \left[ \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right]\) and \(B = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 0 \end{array} \right]\), show that \((AB)^{2} = A^{2}B^{2}\) but \(AB \neq BA\).
- Given \(ABAB = AABB\). Left multiply by \(A^{-1}\), then right multiply by \(B^{-1}\).
Let \(A\) and \(B\) be \(n \times n\) matrices for which \(AB\) is invertible. Show that \(A\) and \(B\) are both invertible.
If \(B\mathbf{x} = \mathbf{0}\) where \(\mathbf{x}\) is \(n \times 1\), then \(AB\mathbf{x} = \mathbf{0}\) so \(\mathbf{x} = \mathbf{0}\) as \(AB\) is invertible. Hence \(B\) is invertible by Theorem [thm:004553], so \(A = (AB)B^{-1}\) is invertible.
Consider \(A = \left[ \begin{array}{rrr} 1 & 3 & -1 \\ 2 & 1 & 5 \\ 1 & -7 & 13 \end{array} \right]\), \(B = \left[ \begin{array}{rrr} 1 & 1 & 2 \\ 3 & 0 & -3 \\ -2 & 5 & 17 \end{array} \right]\).
- [Hint: Row 3 of \(A\) equals \(2\mbox{(row 2) }- 3\mbox{(row 1)}\).]
- [Hint: Column 3 \(= 3\mbox{(column 2) } - \mbox{ column 1}\).]
- \(B \left[ \begin{array}{r} -1 \\ 3 \\ -1 \end{array} \right] = 0\) so \(B\) is not invertible by Theorem [thm:004553].
Show that a square matrix \(A\) is invertible if and only if it can be left-cancelled: \(AB = AC\) implies \(B = C\).
If \(U^{2} = I\), show that \(I + U\) is not invertible unless \(U = I\).
- If \(J\) is the \(4 \times 4\) matrix with every entry \(1\), show that \(I - \frac{1}{2}J\) is self-inverse and symmetric.
- If \(X\) is \(n \times m\) and satisfies \(X^{T}X = I_{m}\), show that \(I_{n} - 2XX^{T}\) is self-inverse and symmetric.
- Write \(U = I_{n} - 2XX^{T}\). Then \(U^{T} = I_{n}^{T} - 2X^{TT}X^{T} = U\), and \(U^{2} = I_{n}^{2} - (2XX^{T})I_{n} - I_{n}(2XX^{T}) + 4(XX^{T})(XX^{T}) = I_{n} - 4XX^{T} + 4XX^{T} = I_{n}\).
An \(n \times n\) matrix \(P\) is called an idempotent if \(P^{2} = P\). Show that:
- \(I\) is the only invertible idempotent.
- \(P\) is an idempotent if and only if \(I - 2P\) is self-inverse.
- \(U\) is self-inverse if and only if \(U = I - 2P\) for some idempotent \(P\).
- \(I - aP\) is invertible for any \(a \neq 1\), and that \((I - aP)^{-1} = I + \left(\frac{a}{1 - a}\right)^{P}\).
- \((I - 2P)^{2} = I - 4P + 4P^{2}\), and this equals \(I\) if and only if \(P^{2} = P\).
If \(A^{2} = kA\), where \(k \neq 0\), show that \(A\) is invertible if and only if \(A = kI\).
Let \(A\) and \(B\) denote \(n \times n\) invertible matrices.
- Show that \(A^{-1} + B^{-1} = A^{-1}(A + B)B^{-1}\).
- If \(A + B\) is also invertible, show that \(A^{-1} + B^{-1}\) is invertible and find a formula for \((A^{-1} + B^{-1})^{-1}\).
- \((A^{-1} + B^{-1})^{-1} = B(A + B)^{-1}A\)
Let \(A\) and \(B\) be \(n \times n\) matrices, and let \(I\) be the \(n \times n\) identity matrix.
- Verify that \(A(I + BA) = (I + AB)A\) and that \((I + BA)B = B(I + AB)\).
- If \(I + AB\) is invertible, verify that \(I + BA\) is also invertible and that \((I + BA)^{-1} = I - B(I + AB)^{-1}A\).