JENECONKMaths Universe

3 × 3 Determinants and Minors

Ages 14–15

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3 × 3 Determinants and Minors

This lesson uses the 2 × 2 determinant rule you learned first. A 3 × 3 matrix has three rows and three columns. To find its determinant, choose one row or column and deal with one entry at a time. For an entry, cross out its row and its column; the four numbers left make a small 2 × 2 matrix. That small determinant is called the minor. Multiply the original entry by its minor, use the alternating signs, and add the three results. A first-row expansion always follows +, −, +. If this feels new, return to the 2 × 2 determinant lesson before continuing.

Your goals

  • ✓You can expand a 3 × 3 determinant using minors and cofactors.
  • ✓You can use minor, cofactor, cofactor expansion correctly.
  • ✓You can write the alternating signs and keep each 2 × 2 minor in brackets.

Words to know

minor:
The determinant left after crossing out an entry's row and column.
cofactor:
A minor with its alternating positive or negative sign.
cofactor expansion:
Adding entries multiplied by their signed minors.

Examples

  • 1For A = 123045106, choose the first row. The three entries are 1, 2 and 3, so the sign pattern is +, −, +.
  • 2For the 1: cross out row 1 and column 1. The small matrix is 4506, whose determinant is 4×6 − 5×0 = 24.
  • 3For the 2: cross out row 1 and column 2. The small matrix is 0516, whose determinant is 0×6 − 5×1 = −5. The middle term is subtracted because its sign is negative.
  • 4For the 3: cross out row 1 and column 3. The small matrix is 0410, whose determinant is 0×0 − 4×1 = −4.
  • 5Now combine the signed terms: det(A) = 1×24 − 2×(−5) + 3×(−4) = 24 + 10 − 12 = 22.
  • 6The sign pattern is a checkerboard: + − + on row 1, − + − on row 2, and + − + on row 3. A zero entry can make one whole term disappear.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Cofactor expansion

Best for: A 3 × 3 determinant
  1. 1.Choose a row or column, preferably with zeros.
  2. 2.Form each 2 × 2 minor by crossing out a row and column.
  3. 3.Apply alternating signs and add the three terms.

Example: Across the first row the signs are +, −, +, so the middle minor is subtracted.

Method 2

Second-route check

Best for: Catching a sign error
  1. 1.Expand along a different row or column.
  2. 2.Recalculate the minors carefully.
  3. 3.Confirm both routes give the same determinant.

Example: A zero in a row can make a second expansion much shorter.

Check these

  • • Do not forget the negative middle cofactor or copy the wrong row and column into a minor.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own 3 × 3 determinants and minors example, solve it, and explain how you checked it.

Byte

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Helpful Hints

“Try drawing a picture to help you see the answer.”