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Inverse Matrices

Inverse Matrices

Definition

  1. An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that $AB=BA=I$. Such a matrix $B$ is unique and called the inverse matrix of $A$, denoted by $A^{-1}$
Summary

Let $A, B$ be $n\times n$ matrices.

  1. $A$ is invertible if and only if $\rref([ A \mid I_n])=[ I_n \mid A’]$ for some $n\times n$ matrix $A’$. In this case, $A’=A^{-1}$.
  2. $A$ is invertible if and only if $A$ is nonsingular.
  3. If $A, B$ are invertible, then $(AB)^{-1}=B^{-1}A^{-1}$
  4. A $2\times 2$ matrix $A=\begin{bmatrix}
    a & b\\
    c& d
    \end{bmatrix}$ is invertible if and only if the determinant $\det(A)=ad-bc \neq 0$.
    If $A$ is invertible, then the inverse matrix is given by $A^{-1}=\frac{1}{\det(A)}\begin{bmatrix}
    d & -b\\
    -c& a
    \end{bmatrix}$.
  5. If $A$ is invertible, then $A^{\trans}$ is invertible and $(A^{\trans})^{-1}=(A^{-1})^{\trans}$.

=solution

Problems

  1. For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.
    (a) $A=\begin{bmatrix}
    1 & 3 & -2 \\
    2 &3 &0 \\
    0 & 1 & -1
    \end{bmatrix}$
    (b) $A=\begin{bmatrix}
    1 & 0 & 2 \\
    -1 &-3 &2 \\
    3 & 6 & -2
    \end{bmatrix}$.

  2. Let A be the matrix
    \[\begin{bmatrix}
    1 & -1 & 0 \\
    0 &1 &-1 \\
    0 & 0 & 1
    \end{bmatrix}.\] Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.
    (The Ohio State University)

  3. Find the inverse matrix of
    \[A=\begin{bmatrix}
    1 & 1 & 2 \\
    0 &0 &1 \\
    1 & 0 & 1
    \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason.
    (The Ohio State University)

  4. Let $A$ be the following $3\times 3$ upper triangular matrix.
    \[A=\begin{bmatrix}
    1 & x & y \\
    0 &1 &z \\
    0 & 0 & 1
    \end{bmatrix},\] where $x, y, z$ are some real numbers. Determine whether the matrix $A$ is invertible or not. If it is invertible, then find the inverse matrix $A^{-1}$.

  5. For which choice(s) of the constant $k$ is the following matrix invertible?
    \[A=\begin{bmatrix}
    1 & 1 & 1 \\
    1 &2 &k \\
    1 & 4 & k^2
    \end{bmatrix}.\] (Johns Hopkins University)

  6. Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.
  7. Let $A$ be an $n \times n$ matrix satisfying $A^2+c_1A+c_0I=O$, where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix. Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular). How about the converse? Namely, is it true that if $c_0=0$, then the matrix $A$ is not invertible?

  8. A square matrix $A$ is called idempotent if $A^2=A$. Show that a square invertible idempotent matrix is the identity matrix.

  9. Find the inverse matrix of $A=\begin{bmatrix}
    1 & 0 & 1 \\
    1 &0 &0 \\
    2 & 1 & 1
    \end{bmatrix}$ if it exists. If you think there is no inverse matrix of $A$, then give a reason.

    See (a)
  10. Find a nonsingular $2\times 2$ matrix $A$ such that $A^3=A^2B-3A^2$, where $B=\begin{bmatrix}
    4 & 1\\
    2& 6
    \end{bmatrix}$. Verify that the matrix $A$ you obtained is actually a nonsingular matrix.

    See (b)
  11. Determine whether there exists a nonsingular matrix $A$ if $A^2=AB+2A$, where $B$ is the following matrix. If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.
    (a) \[B=\begin{bmatrix}
    -1 & 1 & -1 \\
    0 &-1 &0 \\
    1 & 2 & -2
    \end{bmatrix}\] (b) \[B=\begin{bmatrix}
    -1 & 1 & -1 \\
    0 &-1 &0 \\
    2 & 1 & -4
    \end{bmatrix}.\]
  12. Determine whether there exists a nonsingular matrix $A$ if $A^4=ABA^2+2A^3$, where $B$ is the following matrix. $B=\begin{bmatrix}
    -1 & 1 & -1 \\
    0 &-1 &0 \\
    2 & 1 & -4
    \end{bmatrix}.$ If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.
    (The Ohio State University, Linear Algebra Final Exam Problem)

  13. Let $A, B, C$ be the following $3\times 3$ matrices.
    \[A=\begin{bmatrix}
    1 & 2 & 3 \\
    4 &5 &6 \\
    7 & 8 & 9
    \end{bmatrix}, B=\begin{bmatrix}
    1 & 0 & 1 \\
    0 &3 &0 \\
    1 & 0 & 5
    \end{bmatrix}, C=\begin{bmatrix}
    -1 & 0\ & 1 \\
    0 &5 &6 \\
    3 & 0 & 1
    \end{bmatrix}.\] Then compute and simplify the following expression.
    \[(A^{\trans}-B)^{\trans}+C(B^{-1}C)^{-1}.\] (The Ohio State University)

  14. Let $A$ be the coefficient matrix of the system of linear equations
    \begin{align*}
    -x_1-2x_2&=1\\
    2x_1+3x_2&=-1.
    \end{align*}
    (a) Solve the system by finding the inverse matrix $A^{-1}$.
    (b) Let $\mathbf{x}=\begin{bmatrix}
    x_1 \\
    x_2
    \end{bmatrix}$ be the solution of the system obtained in part (a). Calculate and simplify
    \[A^{2017}\mathbf{x}.\] (The Ohio State University)

  15. (a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as $A=BC$, where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix. Prove that the matrix $A$ cannot be invertible.
    (b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be written as $A=BC$, where $B$ is a $ 2\times 3$ matrix and $C$ is a $3\times 2$ matrix. Can the matrix $A$ be invertible?

  16. For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions.
    (1) $A=aP+(a+1)Q$, (2) $P^2=P$, (3) $Q^2=Q$, (4) $PQ=O$, (5) $QP=O$, where $O$ is the $2\times 2$ zero matrix. Then do the following problems.
    (a) Prove that $(P+Q)A=A$.
    (b) Suppose $a$ is a positive real number and let $A=\begin{bmatrix}
    a & 0\\
    1& a+1
    \end{bmatrix}$. Then find all matrices $P, Q$ satisfying conditions (1)-(5).
    (c) Let $n$ be an integer greater than $1$. For any integer $k$, $2\leq k \leq n$, we define the matrix $A_k=\begin{bmatrix}
    k & 0\\
    1& k+1
    \end{bmatrix}$. Then calculate and simplify the matrix product $A_nA_{n-1}A_{n-2}\cdots A_2$.
    (Tokyo University Entrance Exam 2007)

  17. Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by $A=I-a\mathbf{v}\mathbf{v}^{\trans}$, where $I$ is the $n\times n$ identity matrix. Prove that $A$ is a symmetric matrix and $AA=I$. Conclude that the inverse matrix is $A^{-1}=A$.
  18. Consider the system of linear equations
    \begin{align*}
    x_1&= 2, \\
    -2x_1 + x_2 &= 3, \\
    5x_1-4x_2 +x_3 &= 2
    \end{align*}
    (a) Find the coefficient matrix and its inverse matrix.
    (b) Using the inverse matrix, solve the system of linear equations.
    (The Ohio State University)

  19. Consider the following system of linear equations
    \begin{align*}
    2x+3y+z&=-1\\
    3x+3y+z&=1\\
    2x+4y+z&=-2.
    \end{align*}
    (a) Find the coefficient matrix $A$ for this system.
    (b) Find the inverse matrix of the coefficient matrix found in (a)
    (c) Solve the system using the inverse matrix $A^{-1}$.

  20. Consider the matrix
    \[A=\begin{bmatrix}
    1 & 2 & 1 \\
    2 &5 &4 \\
    1 & 1 & 0
    \end{bmatrix}.\] (a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why.
    (b) Are the vectors
    \[ \mathbf{A}_1=\begin{bmatrix}
    1 \\
    2 \\
    1
    \end{bmatrix}, \mathbf{A}_2=\begin{bmatrix}
    2 \\
    5 \\
    1
    \end{bmatrix},
    \text{ and } \mathbf{A}_3=\begin{bmatrix}
    1 \\
    4 \\
    0
    \end{bmatrix}\] linearly independent?
    (c) Write the vector $\mathbf{b}=\begin{bmatrix}
    1 \\
    1 \\
    1
    \end{bmatrix}$ as a linear combination of $\mathbf{A}_1$, $\mathbf{A}_2$, and $\mathbf{A}_3$.
    (The Ohio State University, Linear Algebra Exam)

  21. A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$. Find $\det(A)$.
    See (a)
  22. Suppose that a real matrix $A$ maps each of the following vectors
    \[\mathbf{x}_1=\begin{bmatrix}
    1 \\
    1 \\
    1
    \end{bmatrix}, \mathbf{x}_2=\begin{bmatrix}
    0 \\
    1 \\
    1
    \end{bmatrix}, \mathbf{x}_3=\begin{bmatrix}
    0 \\
    0 \\
    1
    \end{bmatrix} \] into the vectors
    \[\mathbf{y}_1=\begin{bmatrix}
    1 \\
    2 \\
    0
    \end{bmatrix}, \mathbf{y}_2=\begin{bmatrix}
    -1 \\
    0 \\
    3
    \end{bmatrix}, \mathbf{y}_3=\begin{bmatrix}
    3 \\
    1 \\
    1
    \end{bmatrix},\] respectively. That is, $A\mathbf{x}_i=\mathbf{y}_i$ for $i=1,2,3$. Find the matrix $A$.
    (Kyoto University Exam)

  23. Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that
    \[A(A+B)^{-1}B=B(A+B)^{-1}A.\] (University of California, Berkeley Qualifying Exam)

  24. Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques.

  25. Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies
    \[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\] Define the matrix
    \[A=I+\mathbf{u}\mathbf{v}^{\trans}.\] Prove that $A$ is invertible and the inverse matrix is given by the formula
    \[A^{-1}=I-a\mathbf{u}\mathbf{v}^{\trans},\] where
    \[a=\frac{1}{1+\mathbf{v}^{\trans}\mathbf{u}}.\] This formula is called the Sherman-Woodberry formula.

  26. Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
    \[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\] Using the formula, calculate the inverse matrix of $\begin{bmatrix}
    2 & 1\\
    1& 2
    \end{bmatrix}$.

  27. Let $A=\begin{bmatrix}
    a & 0\\
    0& b
    \end{bmatrix}$. Show that
    (a) $A^n=\begin{bmatrix}
    a^n & 0\\
    0& b^n
    \end{bmatrix}$ for any $n \in \N$.
    (b) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix. Show that $B^n=S^{-1}A^n S$ for any $n \in \N$

  28. Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$. Namely, show that $(A^{\trans})^{-1}=(A^{-1})^{\trans}$.
  29. A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.
    (a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent.
    (b) Let $P$ be an invertible $n \times n$ matrix and let $N$ be a nilpotent $n\times n$ matrix. Is the product $PN$ nilpotent? If so, prove it. If not, give a counterexample.

  30. (a) Show that if $A$ is invertible, then $A$ is nonsingular.
    (b) Let $A, B, C$ be $n\times n$ matrices such that $AB=C$. Prove that if either $A$ or $B$ is singular, then so is $C$.
    (c) Show that if $A$ is nonsingular, then $A$ is invertible.

  31. A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix. Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of the same size as $A$. Is the matrix $B-A$ invertible? If so prove it. Otherwise, give a counterexample.
  32. Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be $C_{ij}=(-1)^{ij}\det(M_{ij})$, where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix $C=(C_{ij})$, and define the $n\times n$ matrix $\Adj(A)=C^{\trans}$. The matrix $\Adj(A)$ is called the adjoint matrix of $A$. When $A$ is invertible, then its inverse can be obtained by the formula
    \[A^{-1}=\frac{1}{\det(A)}\Adj(A).\] For each of the following matrices, determine whether it is invertible, and if so, then find the invertible matrix using the above formula.
    (a) $A=\begin{bmatrix}
    1 & 5 & 2 \\
    0 &-1 &2 \\
    0 & 0 & 1
    \end{bmatrix}$.
    (b) $B=\begin{bmatrix}
    1 & 0 & 2 \\
    0 &1 &4 \\
    3 & 0 & 1
    \end{bmatrix}$.

  33. Let $A$ be a real symmetric matrix whose diagonal entries are all positive real numbers. Is it true that the all of the diagonal entries of the inverse matrix $A^{-1}$ are also positive? If so, prove it. Otherwise, give a counterexample.
  34. Let $A$ be an $n\times n$ nonsingular matrix with integer entries. Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.