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• Find All Values of $x$ such that the Matrix is Invertible Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if \[ A= \begin{bmatrix} 1 & 0 & c \\ 0 & a & -b \\ -1/a & x & x^{2} \end{bmatrix} . \]   Solution. We know that $A$ is invertible precisely when […]
• Is the Derivative Linear Transformation Diagonalizable? Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by \[ T( ax^2 + bx + c ) = 2ax + b . \] Is $T$ diagonalizable? If so, find a diagonal matrix which […]
• Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable Determine all eigenvalues and their algebraic multiplicities of the matrix \[A=\begin{bmatrix} 1 & a & 1 \\ a &1 &a \\ 1 & a & 1 \end{bmatrix},\] where $a$ is a real number.   Proof. To find eigenvalues we first compute the characteristic polynomial of the […]
• Ideal Quotient (Colon Ideal) is an Ideal Let $R$ be a commutative ring. Let $S$ be a subset of $R$ and let $I$ be an ideal of $I$. We define the subset \[(I:S):=\{ a \in R \mid aS\subset I\}.\] Prove that $(I:S)$ is an ideal of $R$. This ideal is called the ideal quotient, or colon ideal.   Proof. Let $a, […]
• 10 True or False Problems about Basic Matrix Operations Test your understanding of basic properties of matrix operations. There are 10 True or False Quiz Problems. These 10 problems are very common and essential. So make sure to understand these and don't lose a point if any of these is your exam problems. (These are actual exam […]
• Compute the Determinant of a Magic Square Let \[ A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} . \] Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic […]
• Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism? Let $V$ denote the vector space of all real $2\times 2$ matrices. Suppose that the linear transformation from $V$ to $V$ is given as below. \[T(A)=\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}A-A\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}.\] Prove or […]
• 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? […]