# Find the Rank of a Matrix with a Parameter

## Problem 103

Find the rank of the following real matrix.
$\begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},$ where $a$ is a real number.

(Kyoto University, Linear Algebra Exam)

## Solution.

The rank is the number of nonzero rows of a (reduced) row echelon form matrix of the given matrix.

We apply elementary row operations as follows.
\begin{align*}
&\begin{bmatrix}
a & 1 & 2 \\
1 &1 &1 \\
-1 & 1 & 1-a
\end{bmatrix}
\xrightarrow{R_1 \leftrightarrow R_2}
\begin{bmatrix}
1 &1 &1 \\
a & 1 & 2 \\
-1 & 1 & 1-a
\end{bmatrix}
\xrightarrow[R_3+R_1]{R_2-aR_1}
\begin{bmatrix}
1 &1 &1 \\
0 & 1-a & 2-a \\
0 & 2 & 2-a
\end{bmatrix}\8pt] &\xrightarrow{R_2\leftrightarrow R_3} \begin{bmatrix} 1 &1 &1 \\ 0 & 2 & 2-a\\ 0 & 1-a & 2-a \end{bmatrix} \xrightarrow{R_3-\frac{1-a}{2}R_2} \begin{bmatrix} 1 &1 &1 \\ 0 & 2 & 2-a\\ 0 & 0 & (2-a) (a+1)/2 \end{bmatrix}. \end{align*} The last matrix is in row echelon form. Therefore, if a \neq -1, 2, then (3, 3)-entry of the last matrix is not zero. From this we see that the rank is 3 when a \neq -1, 2. On the other hand, when a=-1 or a=2 the third row is a zero row, hence the rank is 2. Sponsored Links ### More from my site • Find a Matrix that Maps Given Vectors to Given Vectors Suppose that a real matrix A maps each of the following vectors \[\mathbf{x}_1=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} into the […]
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