# Find Values of $a$ so that Augmented Matrix Represents a Consistent System

## Problem 249

Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.

\[A= \left[\begin{array}{rrr|r}

1 & 2 & 3 & 4 \\

2 &-1 & -2 & a^2 \\

-1 & -7 & -11 & a

\end{array} \right],\]
where $a$ is a real number. Determine all the values of $a$ so that the corresponding system is consistent.

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## Solution.

We apply the elementary row operations to $A$ as follows.

\begin{align*}

A= \left[\begin{array}{rrr|r}

1 & 2 & 3 & 4 \\

2 &-1 & -2 & a^2 \\

-1 & -7 & -11 & a

\end{array} \right]
\xrightarrow{\substack{R_2-2R_2\\ R_3+R_1}}

\left[\begin{array}{rrr|r}

1 & 2 & 3 & 4 \\

0 &-5 & -8 & a^2-8 \\

0 & -5 & -8 & a+4

\end{array} \right]\\

\xrightarrow{R_3-R_2}

\left[\begin{array}{rrr|r}

1 & 2 & 3 & 4 \\

0 &-5 & -8 & a^2-8 \\

0 & 0 & 0 & -a^2+a+12

\end{array} \right].

\end{align*}

The last matrix is in row echelon form. (It’s not in reduced row echelon form.)

Then the system is consistent if and only if $-a^2+a+12$ is zero. (See Remark below.)

Since we can factor

\[0=-a^2+a+12=-(a+3)(a-4),\]
we see that the system is consistent if and only if $a=-3, 4$.

### Remark

If $-a^2+a+12 \neq 0$, then we divide the third row by $-a^2+a+12$ and obtain

\[ \left[\begin{array}{rrr|r}

0 & 0 & 0 & 1

\end{array} \right]\]
in the third row.

The corresponding linear equation is

\[0x_1+0x_2+0x_3=1,\]
and clearly, there is no solution to this equation. Hence the system is inconsistent.

On the other hand, if $-a^2+a+12 = 0$, then the system has solutions. (You just need to reduce the above matrix further or use the back substitution.)

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