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Back to Linear Algebra

The Determinant

Not a formula to memorise — the factor by which a matrix scales area in 2D and volume in 3D. Negative means flipped. Zero means squashed flat.

Step 1 of 33Speed

Start with the unit square: î and ĵ, area 1. Any matrix moves it to a parallelogram spanned by its two columns. The determinant is that parallelogram's area — with a sign.

A =

\det\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = (1)(1) - (0)(0) = \mathbf{1}

ad − bc is the area of the parallelogram: the big rectangle (a + b)(c + d) minus the corner pieces that are not inside it.

The Determinant

Not a formula to memorise — the factor by which a matrix scales area in 2D and volume in 3D. Negative means flipped. Zero means squashed flat.

Step 1 of 33Speed

Start with the unit square: î and ĵ, area 1. Any matrix moves it to a parallelogram spanned by its two columns. The determinant is that parallelogram's area — with a sign.

A =

\det\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = (1)(1) - (0)(0) = \mathbf{1}

ad − bc is the area of the parallelogram: the big rectangle (a + b)(c + d) minus the corner pieces that are not inside it.

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The Determinant

See also

The Determinant

See also