A determinant has the value 21. Swap two of its rows and what is the new value? You can say it without expanding again: −21. The seven properties on this page are shortcuts of exactly this kind, and every calculation in the next topic stands on them.
1. Three easy properties
Property (i): if every entry of one row (or column) is zero, the determinant is zero. The reason is plain. Expand along that row and every term carries a factor 0, so every term vanishes.
Property (ii): turn the rows into columns and the columns into rows and the value does not change, |Aᵗ| = |A|. So whatever is said about rows is said about columns too.
Property (iii): interchange two rows (or two columns) and the numerical value stays the same while the sign changes.
2. When two rows are the same
Property (iv): if two rows (or two columns) are identical, the value of the determinant is zero.
The book gives the reason through a row operation. Say r₁ = r₃. Then r₁′ = r₁ − r₃ turns the whole first row into zeros, and this operation leaves the value alone (property vii, coming up). A zero row means a zero value. For columns the same move is c₂′ = c₂ − c₃.
3. Scaling, splitting and adding
Property (v): multiply every entry of one row (or column) by m and the determinant is multiplied by m. If it was D, it becomes mD. Read the other way round, a common factor of one row can be taken outside the determinant.
Property (vi): if every entry of one row (or column) is a sum of two numbers, the determinant can be written as the sum of two determinants. Shown on the first column (rows are separated by ;):
Property (vii): add k times the entries of one or more other rows (or columns) to the entries of a row (or column) and the value is unchanged. The book proves it from (vi), (v) and (iv): the sum splits into two determinants, k comes out of the second, and what is left has two identical lines, so it is zero.
Before matching, run the properties. Here is D = |2, 1, 1 ; 0, 5, −2 ; 1, −3, 4| = 21 from section 5. Press an operation: the entries change and the value is worked out again on the spot.
row operationspress an operation
Does the value change?
The numbers are our own example, for practice.
Match the seven properties with the seven effects. Each time ask one thing: does the value change?
match the propertychip, then box
Does the value change?
Pick up a chip, then drop it in a box. Ask: does the value change?
Right. Next one.
Not yet. If it changes, how: the sign flips, it scales by m, or it splits into two determinants?
All in place. A zero value comes from a zero row or two equal rows. The value holds steady under a transpose and under adding a multiple of another row.
4. Reading the operation notation
In determinant work, a row or column operation is written in a short notation. r₁′ = r₁ − r₂ means subtract the second row from the first and put the result in the first row. c₃′ = c₁ + c₂ + c₃ means add the three columns and put the result in the third column. The r is a row, the c is a column, and the r₁, r₂ on the right are the old rows.
read the notation
r is row, c is column
Which sentence does r₁′ = r₁ − r₂ say?
That is c₂′ = c₂ − c₃. Our notation has an r, so it is rows, not columns.
Right. New row 1 = old row 1 − row 2, and row 2 stays as it was.
That is c₃′ = c₁ + c₂ + c₃: columns, and a sum, not a difference.
That is r₂′ = r₂ − r₃. There row 2 is the one replaced, but in ours it is row 1.
5. One determinant, many properties
Take D = |2, 1, 1 ; 0, 5, −2 ; 1, −3, 4| = 21. You can expand it yourself to check. Now pick one operation and type the new value, without expanding again. For a negative, use the keyboard hyphen (-). None of this goes anywhere. The notebook is yours.
what will the value be?pick, then type
New value?
The numbers are our own example, for practice.
One place is easy to muddle. Multiplying one row of a determinant by m multiplies its value by m, but the scalar multiple kA of a matrix multiplies every entry by k. And these seven properties are about the value of a determinant, not of a matrix: a swap in a matrix has no value to flip, so keep the word “determinant” in what you write.
6. Check yourself
Three questions in the exam's style. The first.
check yourselfpick one answer
Question 1
If A = [[1, 1, 1], [e, π, √3], [3, 3, 3]], then |A| = ?
The third row is 3 × the first, so factoring out 3 leaves two identical rows and the value is 0, not e.
For the same reason the value is 0; π is just one entry.
Expanding does not give this expression; with proportional rows the value is 0.
Right. The third row (3, 3, 3) is 3 × (1, 1, 1), a multiple of the first; so |A| = 3 × 0 = 0.
For the second, decide first which of the seven properties it needs.