Topic 5 of 8

Determinants

Step 19 · Linear algebra topics

Compute determinants of 2 × 2 and 3 × 3 matrices, use their properties under row operations and products, and read a determinant as an area or volume scale factor.

Computing determinants

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

For a 3 × 3 matrix, expand along a row: each entry times the determinant of the 2 × 2 matrix left when you cross out that entry's row and column, with the signs alternating +, −, +.

Example: 2 × 2

Find det $\begin{pmatrix} 3 & 5 \\ 2 & 4 \end{pmatrix}$ .

3 × 4 − 5 × 2 = 12 − 10.

Answer2

Example: 3 × 3

Find det $\begin{pmatrix} 2 & 0 & 1 \\ 1 & 3 & 2 \\ 1 & 1 & 2 \end{pmatrix}$ .

Along the first row: 2(3 × 2 − 2 × 1) − 0 + 1(1 × 1 − 3 × 1).
That is 2 × 4 + 1 × (−2).

Answer6

Properties of determinants

Change	Effect on det
Swap two rows	Changes sign
Multiply a row by k	Multiplies by k
Add a multiple of one row to another	No change
Multiply two matrices	det(AB) = det(A) det(B)
Transpose	$\det(A^{T}) = \det(A)$

Example: Using row operations

det A = 5. What is the determinant after swapping two rows and then doubling one row?

Swapping gives −5.
Doubling a row gives −10.

Answer−10

Example: Invertibility

For which k is $\begin{pmatrix} 1 & 2 \\ 3 & k \end{pmatrix}$ not invertible?

det = k − 6.
It is 0 when k = 6.

Answerk = 6

Determinants and area

|det A| is the factor by which A scales areas in the plane, or volumes in space. The sign says whether A keeps or flips orientation, as a mirror does.

The matrix with columns ⟨2, 1⟩ and ⟨1, 3⟩ turns the unit square into a parallelogram of area 5, its determinant.

Example: A parallelogram

Find the area of the parallelogram with sides ⟨2, 1⟩ and ⟨1, 3⟩.

The determinant with those columns is 2 × 3 − 1 × 1 = 5.

Answer5

Example: A volume

A 3 × 3 matrix has determinant −4. What happens to a cube of volume 2?

Volumes are multiplied by 4, the size of −4.
The minus sign means orientation flips, like a mirror image.

AnswerIt becomes a mirror-imaged solid of volume 8

Try it yourself

Work each one on paper first, then open the answer to check.

Find det $\begin{pmatrix} 5 & 2 \\ 3 & 1 \end{pmatrix}$ .
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−1
Find det $\begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}$ .
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24
det A = 3 for a 2 × 2 matrix A. What is det(2A)?
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12
det A = 4 and det B = −2. What is det(AB)?
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−8
Is $\begin{pmatrix} 2 & 6 \\ 1 & 3 \end{pmatrix}$ invertible?
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No: its determinant is 0
Find the area of the triangle with sides ⟨4, 0⟩ and ⟨1, 3⟩ from the origin.
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6