Linear Algebra

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

Problem 138

Find the determinant of the matix
\[A=\begin{bmatrix}
100 & 101 & 102 \\
101 &102 &103 \\
102 & 103 & 104
\end{bmatrix}.\]

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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.

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