# Tagged: Ohio State

## Problem 66

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)

## Problem 65

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, Linear Algebra Exam)

## Problem 47

Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$.
Calculate and simplify the expression
$-T^3+4T^2+5T-2I,$ where $I$ is the $3\times 3$ identity matrix.

(The Ohio State University Linear Algebra Exam)

## Problem 5

Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.
Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.

(The Ohio State University Linear Algebra Exam)