## Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations

## 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*)