# How to Find the Determinant of the $3\times 3$ Matrix

## Problem 138

Find the determinant of the matix

\[A=\begin{bmatrix}

100 & 101 & 102 \\

101 &102 &103 \\

102 & 103 & 104

\end{bmatrix}.\]

## Solution.

Note that the determinant does not change if the $i$-th row is added by a scalar multiple of the $j$-th row if $i \neq j$.

We use this fact about the determinant and compute $\det(A)$ as follows.

\begin{align*}

\det(A)&=\begin{vmatrix}

100 & 101 & 102 \\

101 &102 &103 \\

102 & 103 & 104

\end{vmatrix}\\[5 pt]
&=\begin{vmatrix}

100 & 101 & 102 \\

101 &102 &103 \\

1 & 1 & 1

\end{vmatrix}

\quad (\text{by } R_3-R_2)\\[5 pt]
&=\begin{vmatrix}

100 & 101 & 102 \\

1 &1 &1 \\

1 & 1 & 1

\end{vmatrix}

\quad (\text{by } R_2-R_1)\\[5 pt]
&=\begin{vmatrix}

100 & 101 & 102 \\

1 &1 &1 \\

0 & 0 & 0

\end{vmatrix}

\quad (\text{by } R_3-R_1)\\[5 pt]
&=0 \quad (\text{by the third row cofactor expansion}.)

\end{align*}

Therefore the determinant $\det(A)$ is zero.

Add to solve later

Sponsored Links