Determinants of 3×3 Matrices · Linear Algebra

Practice

Lesson goal

compute a $3\times3$ determinant with the rule of Sarrus;
recognise the six terms and which of them are subtracted;
see that each term takes one entry from every row and every column;
use the shortcut for triangular matrices.

From Two Terms to Six

A 2 by 2 determinant has two terms: the product along the main diagonal minus the product along the other diagonal. A 3 by 3 determinant has six terms: three are added and three are subtracted.

\begin{aligned}\det A&=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}\\&\quad-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}\end{aligned}

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How many terms does the formula for a $3\times3$ determinant have?

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The Rule of Sarrus

Write the first two columns again to the right of the matrix. The three diagonals running down to the right give the added products, and the three running up to the right give the subtracted ones.

\begin{array}{ccc|cc}a&b&c&a&b\\d&e&f&d&e\\g&h&i&g&h\end{array}

\det=aei+bfg+cdh-ceg-afh-bdi

Example

\begin{vmatrix}2&1&3\\0&4&-1\\5&-2&1\end{vmatrix}

Added: $2\cdot4\cdot1=8$ , $1\cdot(-1)\cdot5=-5$ and , which total .

Subtracted: $3\cdot4\cdot5=60$ , $2\cdot(-1)\cdot(-2)=4$ and , which total .

\det=3-64=-61

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Compute $\begin{vmatrix}1&2&0\\3&1&2\\0&1&1\end{vmatrix}$ .

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One From Each Row and Each Column

Follow a diagonal of the rule of Sarrus back into the original matrix: it visits every row once and every column once. That is true of all six terms, and it is the pattern that carries over to larger determinants.

a_{13}a_{21}a_{32}:\ \text{rows }1,2,3,\ \text{columns }3,1,2

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The third added product of the rule of Sarrus starts at $a_{13}$ . Select its three entries in $A$ .

$A=\begin{pmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{pmatrix}$

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Which product can be a term of a $3\times3$ determinant?

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Which product is subtracted in the formula for a $3\times3$ determinant?

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Compute $\begin{vmatrix}2&-1&0\\1&3&-2\\0&4&1\end{vmatrix}$ .

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Triangular Matrices

If a matrix is triangular, five of the six terms contain a zero, because each of them takes an entry from below or above the diagonal. Only the diagonal product survives.

\begin{vmatrix}2&7&-1\\0&3&5\\0&0&4\end{vmatrix}=2\cdot3\cdot4=24

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Compute $\det\begin{pmatrix}3&5&-2\\0&-1&4\\0&0&2\end{pmatrix}$ .

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A $3\times3$ matrix with a row of zeros has determinant $0$ .

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The Rule Stops at Three

The rule of Sarrus is a picture of the six terms of a $3\times3$ determinant, and nothing more. A $4\times4$ determinant has $24$ terms, and its diagonals, however they are drawn, give only $8$ of them.

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The rule of Sarrus, with three columns copied instead of two, computes the determinant of a $4\times4$ matrix.

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For which $x$ is $\begin{vmatrix}x&0&1\\0&2&0\\1&0&1\end{vmatrix}=4$ ?

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What You Learned

A $3\times3$ determinant has six terms: three added, three subtracted.
The rule of Sarrus finds them: copy two columns, add the diagonals going down to the right, subtract those going up.
Each term takes one entry from every row and every column.
A triangular matrix has determinant equal to the product of its diagonal.
The rule of Sarrus works for $3\times3$ matrices only.

Final checkpoint

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Compute $\begin{vmatrix}1&0&2\\2&1&0\\0&3&1\end{vmatrix}$ .

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What is $\det\begin{pmatrix}2&0&0\\0&-3&0\\0&0&4\end{pmatrix}$ ?

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Every term of a $3\times3$ determinant contains exactly one entry from each column.

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Which expression equals $\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}$ ?

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For which $x$ is $\det\begin{pmatrix}2&0&0\\1&x&0\\5&3&1\end{pmatrix}=10$ ?

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Completion

Lesson complete

Great work! You now know how to:

compute a $3\times3$ determinant with the rule of Sarrus;
tell which of the six terms are subtracted;
recognise a term: one entry from every row and every column;
use the product of the diagonal for a triangular matrix.

Mapping the lesson.

Practice

Lesson goal

From Two Terms to Six

The Rule of Sarrus

Example

One From Each Row and Each Column

Triangular Matrices

The Rule Stops at Three

What You Learned

Final checkpoint

Completion

Lesson complete