Determinant/Examples/Order 2

Example of Determinant

Let $\mathbf A = \sqbrk a_2$ be a square matrix of order $2$.

That is, let:

$\mathbf A = \begin {bmatrix} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end {bmatrix}$

Then the determinant of $\mathbf A$ is defined as:

\(\ds \begin {vmatrix} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end{vmatrix}\)

\(=\)

\(\ds \map \sgn {1, 2} a_{1 1} a_{2 2} + \map \sgn {2, 1} a_{1 2} a_{2 1}\)

\(\ds \)

\(=\)

\(\ds a_{1 1} a_{2 2} - a_{1 2} a_{2 1}\)

where $\sgn$ denotes the sign of the permutation.

Sources

• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.6$ Determinant and trace
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): determinant
• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.5$: Determinants
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): determinant
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): determinant