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Introduction to Linear Algebra at the Ohio State University quiz problems and solutions


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  • Equivalent Conditions to be a Unitary MatrixEquivalent Conditions to be a Unitary Matrix A complex matrix is called unitary if $\overline{A}^{\trans} A=I$. The inner product $(\mathbf{x}, \mathbf{y})$ of complex vector $\mathbf{x}$, $\mathbf{y}$ is defined by $(\mathbf{x}, \mathbf{y}):=\overline{\mathbf{x}}^{\trans} \mathbf{y}$. The length of a complex vector […]
  • Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$ Let $S$ be the following subset of the 3-dimensional vector space $\R^3$. \[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \] where $\Z$ is the set of all integers. […]
  • If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is OrthogonalIf $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal Let $A$ be an $n\times n$ real skew-symmetric matrix. (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular. (b) Prove that \[B=(I-A)(I+A)^{-1}\] is an orthogonal matrix.   Proof. (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular. The […]
  • The Order of $ab$ and $ba$ in a Group are the SameThe Order of $ab$ and $ba$ in a Group are the Same Let $G$ be a finite group. Let $a, b$ be elements of $G$. Prove that the order of $ab$ is equal to the order of $ba$. (Of course do not assume that $G$ is an abelian group.)   Proof. Let $n$ and $m$ be the order of $ab$ and $ba$, respectively. That is, \[(ab)^n=e, […]
  • Diagonalize the Upper Triangular Matrix and Find the Power of the MatrixDiagonalize the Upper Triangular Matrix and Find the Power of the Matrix Consider the $2\times 2$ complex matrix \[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$. (d) Using the result of the […]
  • The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers.The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers. Decipher the trick of the following mathematical magic.   The Rule of the Game Here is the game. Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below. For example, let's say I have chosen the numbers $7, 5, 3, 2, […]
  • Similar Matrices Have the Same EigenvaluesSimilar Matrices Have the Same Eigenvalues Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same. Proof. We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]
  • If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent?If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent? Consider the following system of linear equations: \begin{align*} ax_1+bx_2 &=c\\ dx_1+ex_2 &=f\\ gx_1+hx_2 &=i. \end{align*} (a) Write down the augmented matrix. (b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? […]

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