MAT 242 Chap.5 Determinants by Expansion by Minors
Determinants by Expansion by Minors
A number that decides invertibility
For a 2 × 2 matrix, ad − bc tells you whether an inverse exists. The determinant generalises that number to every square matrix: det A is nonzero exactly when A is invertible.
The course computes it by expansion by minors, and tests expect the complete expansion to be written out.
Minors, cofactors and the checkerboard
The minor of an entry is the determinant of what remains after deleting that entry's row and column. The cofactor adds a sign from a checkerboard pattern that begins with plus in the top-left corner.
The determinant is the sum of any row's entries times their cofactors, and the same value comes from any row or any column.
That freedom is the whole strategy, because the work grows explosively with size; choosing the row or column with the most zeros can reduce a four-by-four to a single three-by-three.
Shortcuts and properties
Sarrus's diagonal method is allowed for 3 × 3 matrices and fails for anything larger.
Determinants multiply across products, powers carry through, the determinant of an inverse is the reciprocal, and scaling an n × n matrix by r multiplies its determinant by r to the power n. Transposing leaves the value unchanged, and a triangular matrix has determinant equal to the product of its diagonal.
The paid chapter works a sparse four-by-four, cross-checks a three-by-three by Sarrus, and finds the values of a parameter that make a determinant vanish.
Signs are the main hazard: an expansion down a column follows that column's checkerboard pattern, which does not always start with plus.
What the course asks for on tests
The Test 1 skills list asks for determinants by expansion by minors and possibly Sarrus's method.
The course notes allow one shortcut on tests: a term multiplied by zero may be written without evaluating its minor, but the full expansion, including those zero terms, still has to appear. A determinant with a parameter is solved the same way, followed by setting the resulting polynomial equal to zero and factoring.
Expanding along a row that is almost entirely zero, then using Sarrus's method on a remaining three-by-three, is the fastest reliable route through a four-by-four.
The transpose and triangular matrices
Transposing swaps rows and columns, turning an m × n matrix into an n × m one. It leaves a determinant unchanged, which is why expanding down a column is as valid as expanding along a row.
For a triangular matrix every expansion down the first column leaves a single term, so the determinant is the product of the diagonal entries; that shortcut also explains why triangular matrices show their eigenvalues on the diagonal later on. The same observation gives a fast check on any hand expansion of a triangular block.
What this chapter covers
- 01
Minors and cofactors
- 02
The checkerboard of signs
- 03
Expanding along the sparsest row or column
- 04
Sarrus's method for 3 × 3 matrices
- 05
Properties of determinants
Worked example · free
A 3 × 3 determinant by cofactor expansion
- 3Expand along row 1, which contains a zero: 5·(4·6 − 0·(−2)) − 0 + 2·(3·(−2) − 4·0).
- 2Evaluate: 5·24 + 2·(−6) = 108.
Key terms
- Minor
- A minor is the determinant of the submatrix left after deleting one row and one column.
- Cofactor
- A cofactor is a minor multiplied by the sign (−1) to the power i + j from the checkerboard pattern.
- Transpose
- The transpose turns rows into columns, so entry (i, j) of Aᵀ is entry (j, i) of A.
Determinants by Expansion by Minors FAQ
Can Sarrus's method be used on a 4 × 4 matrix?
No. The diagonal shortcut is only valid for three-by-three matrices. For a four-by-four, expand by minors along a convenient row or column and then use Sarrus on the three-by-three minors if you like.
What happens to the determinant when the whole matrix is doubled?
Each of the n rows doubles, so the determinant is multiplied by 2 to the power n. For a three-by-three matrix that is a factor of eight, not two.
Why expand along a row or column with many zeros?
Each zero entry removes a whole minor from the work. A row with one nonzero entry turns a four-by-four determinant into a single three-by-three calculation, which cuts both time and errors.
Exam move
Practise sign patterns separately from arithmetic: write the checkerboard for rows and columns of a four-by-four until you can do it instantly, then time full expansions. Keep a list of the properties and test each on a two-by-two example you make up; seeing det(2A) equal four times det A once is more memorable than the formula.
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