# Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable

## Problem 482

For which values of constants $a, b$ and $c$ is the matrix
$A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}$ diagonalizable?

(The Ohio State University, Linear Algebra Final Exam Problem)

## Solution.

Note that the matrix $A$ is an upper triangular matrix.
Hence the eigenvalues of $A$ are diagonal entries $7, 2, 3$.

So the $3\times 3$ matrix $A$ has three distinct eigenvalues.
This implies that $A$ is diagonalizable.

Hence, regardless of the values of $a, b, c$, the matrix $A$ is always diagonalizable.
Thus, $a, b, c$ can take arbitrary values.

## Final Exam Problems and Solution. (Linear Algebra Math 2568 at the Ohio State University)

This problem is one of the final exam problems of Linear Algebra course at the Ohio State University (Math 2568).

The other problems can be found from the links below.

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