Determinant

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Determinant: meaning, method and solved example (Precalculus) – Maths PDF

In short: The determinant squeezes a square grid of numbers into one value, ad – bc in the 2 × 2 case, and that value reveals at a glance whether the grid can be reversed.

PrecalculusAlgebraN-VM.C.10

What is Determinant?

The determinant boils a square grid of numbers down to one value that carries a lot of information. If it is zero, the matrix squashes the plane flat and cannot be undone; if it is not zero, the matrix is invertible.

Starting from \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), the rule is \(\det A = ad – bc\). Geometrically, its absolute value is the area of the parallelogram spanned by the two column vectors, and a negative sign means the orientation is reversed. For larger matrices it is found by expanding along a row or column.

How to spot it

  • A question asks whether a square matrix has an inverse, or whether a system has a single solution.
  • Vertical bars surround a grid of numbers, or the abbreviation det appears.
  • The area of a parallelogram or triangle is requested from coordinates of vectors.
  • A formula for the inverse of a 2 × 2 matrix divides by a number built from its entries.

The method

  1. Confirm that the matrix is square, since a determinant exists only for equal rows and columns.
  2. For 2 × 2, multiply the main diagonal and subtract the product of the other diagonal: \(ad – bc\).
  3. For 3 × 3, expand along a row, using alternating signs \(+, -, +\) times each entry and its 2 × 2 minor.
  4. Interpret: zero means no inverse; otherwise the absolute value is an area (or volume) scale factor.
⟨4, 2⟩⟨-3, 5⟩area 26det [4 -3; 2 5]= 4·5 – (-3)·2 = 26
The columns ⟨4, 2⟩ and ⟨-3, 5⟩ span a parallelogram of area 26 = det.

Determinant: solved example

Problem

Evaluate \(\det\begin{bmatrix} 4 & -3 \\ 2 & 5 \end{bmatrix}\), then interpret the answer as the area of a parallelogram built from the two columns.

Solution

Apply \(ad – bc\): \(4(5) – (-3)(2) = 20 + 6 = 26\).

It is positive and nonzero, so the matrix is invertible, and the parallelogram with sides \(\langle 4, 2\rangle\) and \(\langle -3, 5\rangle\) has area 26 square units.

Try it

Question 1

Find \(\det\begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix}\). Is the matrix invertible?

Show the answer

\(6(3) – 9(2) = 18 – 18 = 0\). No inverse exists.

Question 2

Find \(\det\begin{bmatrix} 1 & 2 & 0 \\ 3 & -1 & 4 \\ 2 & 5 & -2 \end{bmatrix}\).

Show the answer

Expand along row 1: \(1(2 – 20) – 2(-6 – 8) + 0 = -18 + 28 = 10\).

Determinant vs. Matrix

A matrix is the full rectangular array, and it can have any number of rows and columns. The determinant is just one number that can be extracted from a square array. Different matrices can share a determinant, so the number cannot be turned back into the grid.

Matrix

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