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Pre-Calculus Matrices

Determinants of 2×2 and 3×3 matrices

20 practice questions 0 video lessons Theory + worked examples

Determinants

Common Core Pre-Calculus • Standard N-VM.12 • Matrices

Determinants is a topic in Matrices in the Common Core State Standards. It is aligned to Standard N-VM.12, which requires students to compute the determinant and use it to decide whether a matrix has an inverse.

The determinant of a \(2\times 2\) matrix is \(ad-bc\); a nonzero determinant means the matrix is invertible, and its absolute value is the area scale factor.

Common Core Pre-Calculus › Matrices › Determinants  —  Standard N-VM.12

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Theory

The determinant is a single number computed from a square matrix. For a \(2\times 2\):

\[\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc.\]

A \(3\times 3\) determinant is found by cofactor expansion along a row or column. The determinant tells you two important things:

  • Invertibility: \(\det\neq 0\) means the matrix has an inverse; \(\det=0\) means it is singular.
  • Area/volume scaling: \(|\det|\) is the factor by which the matrix scales area (2D) or volume (3D).
\(\det=0\) is the warning sign: no inverse, and the transformation collapses area to zero.
2×2 determinant 2×2 determinant 2×2 determinant det [a b; c d] = ad − bc
The \(2\times 2\) determinant \(ad-bc\).
What the determinant tells you What the determinant tells you What the determinant tells you det ≠ 0: invertible det = 0: singular (no inverse) |det| = area scale factor
What the determinant tells you.

The 2×2 determinant and a cofactor expansion for 3×3:

\[\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc\]
\[\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\]
2 by 2 determinant is a d minus b c; the 3 by 3 uses cofactor expansion
Watch the sign pattern \(+\,-\,+\) across the top row of a cofactor expansion.

How to compute a determinant

  1. 2×2: \(ad-bc\).
  2. 3×3: expand along a row/column with the \(+\,-\,+\) signs, using \(2\times 2\) minors.
  3. Interpret: \(\det=0\) means singular; \(|\det|\) is the area scale.
Example 1 — A 2×2 determinant
Find \(\det\begin{bmatrix}3&5\\2&4\end{bmatrix}\).
Solution

Use \(ad-bc\).

\(\det\)\(=\)\((3)(4)-(5)(2)\)
\(=\)\(12-10=2\)
determinant is 2
Example 2 — A singular matrix
Find \(\det\begin{bmatrix}2&4\\1&2\end{bmatrix}\) and say whether it is invertible.
Solution

Compute the determinant.

\(\det\)\(=\)\((2)(2)-(4)(1)=0\)

Determinant \(0\) \(\Rightarrow\) not invertible (singular).

determinant is 0, so the matrix is not invertible
Example 3 — A 3×3 determinant
Find \(\det\begin{bmatrix}1&2&3\\0&1&4\\0&0&2\end{bmatrix}\).
Solution

Expand along the first column (only the top entry is nonzero); for this upper-triangular matrix the determinant is the product of the diagonal.

\(\det\)\(=\)\(1\cdot 1\cdot 2=2\)
determinant is 2, the product of the diagonal
Example 4 — Area interpretation
What does \(\det\begin{bmatrix}3&0\\0&2\end{bmatrix}=6\) mean geometrically?
Solution

The matrix scales the plane, and \(|\det|=6\) is the factor by which it scales area.

|\det|\(=\)\(6\)

A unit square maps to a region of area 6.

the absolute value of the determinant is the area scale factor, 6

Common pitfalls

\(ad-bc\), not \(ab-cd\). Multiply the diagonals, then subtract.
Alternate the signs in a cofactor expansion: \(+,-,+\).
\(\det=0\) means no inverse. Don't try to invert a singular matrix.

Frequently asked questions

How do you find a 2x2 determinant?

\(\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc\): the product of the main diagonal minus the product of the other diagonal.

What does the determinant tell you?

Whether the matrix is invertible (\(\det\neq 0\)) and, in absolute value, the factor by which it scales area or volume.

What is a singular matrix?

One with determinant \(0\); it has no inverse and collapses area to zero.

How do you find a 3x3 determinant?

Use cofactor expansion along a row or column, alternating signs \(+,-,+\) and using \(2\times 2\) minors.

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