Concept 03 / 09Matrices and Determinants
Determinants: Definition, Expansion and Geometric Meaning
The 2 x 2 and 3 x 3 determinants, expansion along any row or column with the checkerboard of signs, the determinant as a signed area or volume scale factor, and why det(AB) = det A det B while det(A + B) has no such rule.
To every square matrix we attach a number, its determinant, written or . For a matrix,
.
For a matrix, expand along the first row, multiplying each entry by the determinant left after deleting its row and column, with alternating signs:
.
You may expand along any row or any column and get the same value, provided you use the checkerboard of signs: the sign attached to position is , giving in the first row, in the second, and in the third. In practice, expand along the row or column with the most zeros.
Minors and cofactors. The minor is the determinant obtained by deleting row and column . The cofactor is . Expansion along row reads . A striking fact used to derive the inverse: if you multiply the entries of one row by the cofactors of a different row, the sum is , because that sum is the determinant of a matrix with two equal rows.
Geometric meaning. The two columns and of a matrix span a parallelogram, and is its area with a sign: positive if the second column is anticlockwise from the first, negative if clockwise. The figure shows columns and , with area . In three dimensions, the three columns span a parallelepiped, and is its signed volume, which is the scalar triple product of the columns. Viewed as a transformation, multiplies every area (or volume) by , and a negative determinant means the transformation reverses orientation, like a reflection. This single idea explains the main facts.
exactly when the columns are linearly dependent (parallel in 2D, coplanar in 3D): the transformation squashes space into a lower dimension, so no inverse can exist.
: scaling by one factor and then another multiplies the factors. Consequently and .
for an matrix, because each of the directions is stretched by . So for matrices, not .
in general. There is no addition rule, since sums of transformations do not correspond to anything simple about volumes.
Triangular matrices. The determinant of an upper or lower triangular matrix (including a diagonal matrix) is the product of its diagonal entries. Expanding down the first column repeatedly shows this. It is also why row-reduction to triangular form is an efficient way to evaluate large determinants.
Area of a triangle. With vertices , the area is
,
and the three points are collinear exactly when this determinant is . When a question gives the area and asks for an unknown coordinate, remember the absolute value: the determinant equals area, so both signs must be considered.
Key Takeaways:
- : . : expand along any row or column with signs .
- ; row own cofactors ; row another row's cofactors .
- is the area or volume scale factor; its sign records orientation.
- ; ; .
- has no general formula.
- Triangular matrices: the determinant is the product of the diagonal entries.
- Triangle area ; both signs must be considered when solving for unknowns.
- Pitfall: for a matrix. The correct value is .
Brain-Teaser 1: is with . Find , , and .
Solution: . . . . Each follows from 'determinant = volume scale factor': doubling every side of a box multiplies its volume by , and reversing all three axes flips the orientation once overall (three flips).
Brain-Teaser 2: Without expanding fully, show that .
Solution: View the determinant as a polynomial in . If , two rows are equal and , so is a factor. By the same argument, and are factors. has total degree in , and so does the product of the three factors, so for a constant . To find , substitute convenient values: set : and , so . This is the Vandermonde determinant, and the 'equal rows give zero, so find the factors' method is the most efficient way to evaluate symmetric determinants.
Brain-Teaser 3: The points , and are collinear. Find .
Solution: Set the area determinant to zero: . Subtract row 1 from rows 2 and 3 to create zeros in the last column: row 2 becomes and row 3 becomes . Expanding along the last column, the determinant equals . The second product is . So the determinant is . Setting it to gives or . Row operations before expanding keep the arithmetic manageable.
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