# nonsingular matrix

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- The Matrix Exponential of a Diagonal Matrix For a square matrix $M$, its matrix exponential is defined by \[e^M = \sum_{i=0}^\infty \frac{M^k}{k!}.\] Suppose that $M$ is a diagonal matrix \[ M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & […]
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- Find the Matrix Representation of $T(f)(x) = f(x^2)$ if it is a Linear Transformation For an integer $n > 0$, let $\mathrm{P}_n$ denote the vector space of polynomials with real coefficients of degree $2$ or less. Define the map $T : \mathrm{P}_2 \rightarrow \mathrm{P}_4$ by \[ T(f)(x) = f(x^2).\] Determine if $T$ is a linear transformation. If it is, find […]
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- Is the Map $T (f) (x) = f(x) – x – 1$ a Linear Transformation between Vector Spaces of Polynomials? Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_4 \rightarrow \mathrm{P}_{4}$ be the map defined by, for $f \in \mathrm{P}_4$, \[ […]
- For What Values of $a$, Is the Matrix Nonsingular? Determine the values of a real number $a$ such that the matrix \[A=\begin{bmatrix} 3 & 0 & a \\ 2 &3 &0 \\ 0 & 18a & a+1 \end{bmatrix}\] is nonsingular. Solution. We apply elementary row operations and obtain: \begin{align*} A=\begin{bmatrix} 3 & 0 & a […]
- Determine whether the Given 3 by 3 Matrices are Nonsingular Determine whether the following matrices are nonsingular or not. (a) $A=\begin{bmatrix} 1 & 0 & 1 \\ 2 &1 &2 \\ 1 & 0 & -1 \end{bmatrix}$. (b) $B=\begin{bmatrix} 2 & 1 & 2 \\ 1 &0 &1 \\ 4 & 1 & 4 \end{bmatrix}$. Solution. Recall that […]
- Solving a System of Differential Equation by Finding Eigenvalues and Eigenvectors Consider the system of differential equations \begin{align*} \frac{\mathrm{d} x_1(t)}{\mathrm{d}t} & = 2 x_1(t) -x_2(t) -x_3(t)\\ \frac{\mathrm{d}x_2(t)}{\mathrm{d}t} & = -x_1(t)+2x_2(t) -x_3(t)\\ \frac{\mathrm{d}x_3(t)}{\mathrm{d}t} & = -x_1(t) -x_2(t) […]