Properties of determinants

What you will be able to do

• state the properties of determinants and the reason for each;
• reduce a determinant to simple form with row and column operations;
• recognise when a determinant is $0$ without expanding;
• use $|kA| = k^n|A|$ and $|AB| = |A||B|$.

The main properties

Here $A$ is a square matrix of order $n$. Every statement about rows is also true for columns.

1. Transpose. $|A^T| = |A|$.
2. Swap. Interchanging two rows (or two columns) changes the sign of the determinant.
3. Equal rows. If two rows are equal, $|A| = 0$. (Swapping them changes nothing, but must change the sign, so $|A| = -|A|$.)
4. Multiply a row. Multiplying one row by $k$ multiplies $|A|$ by $k$. So a common factor of one row can be taken out of the determinant.
5. Proportional rows. If one row is a multiple of another, $|A| = 0$. If a row is all zeros, $|A| = 0$.
6. Add a multiple. Replacing $R_i$ by $R_i + kR_j$ does not change $|A|$.
7. Scalar multiple of a matrix. $|kA| = k^n|A|$, because $kA$ multiplies all $n$ rows by $k$.
8. Product. $|AB| = |A||B|$.

Property 6 is the useful one. Think of the parallelogram. Adding a multiple of one column to the other slants it, but the base and the height stay the same. So the area stays the same.

Making zeros

The plan: use property 6 to make a row or column full of zeros except one entry. Then expand along it.

Evaluate $

$.

1. Clear column 1. Replace $R_2$ by $R_2 - 2R_1$ and $R_3$ by $R_3 - 3R_1$. The determinant does not change.
2. The new rows. $R_2 = (0, 1, 3)$ and $R_3 = (0, -1, 6)$.
3. Expand along column 1. The value is $2 \cdot

= 2(6 + 3) = 18$.

Recognising a zero

You can often see that a determinant is $0$ before you calculate. Look at this one.

Its rows look innocent. But $R_3 - R_2 = (3, 3, 3)$ and $R_2 - R_1 = (3, 3, 3)$ are equal, so $R_3 - 2R_2 + R_1 = 0$. Replacing $R_3$ by $R_3 - 2R_2 + R_1$ is made of operations of type 6, and it gives a row of zeros. The determinant is $0$.

Another example is the determinant below.

Add column $2$ to column $3$. Every entry of the new column $3$ is $a + b + c$, a common factor. The columns are then proportional to column $1$, so the determinant is $0$.

Factorising

Properties 4 and 6 also factorise determinants. A well-known case is

Subtracting row $1$ from the other two rows gives factors $(b - a)$ and $(c - a)$. For $a = 1$, $b = 2$, $c = 4$ this is $1 \cdot 3 \cdot 2 = 6$.

A second case is the determinant below.

Add columns $2$ and $3$ to column $1$. Column $1$ becomes $(x + 2)$ in every row. Taking out the factor gives

Then $R_2 - R_1$ and $R_3 - R_1$ finish it: the determinant is $(x + 2)(x - 1)^2$.

Scalar multiples and products

A $3 \times 3$ matrix $A$ has $|A| = 4$. Then $|2A| = 2^3 \cdot 4 = 32$. The factor is $2^3$ and not $2$, because every one of the three rows is doubled.

For products, take the two matrices below.

Then $|A| = -2$ and $|B| = -5$. We saw in the lesson on products that

and $|AB| = 110 - 100 = 10 = (-2)(-5)$.

A worked example

A $3 \times 3$ matrix $A$ has $|A| = -2$. Find $|3A|$, $|A^2|$ and $|A^TA|$.

1. $|3A|$. $3^3 \cdot (-2) = -54$.
2. $|A^2|$. $|A||A| = 4$.
3. $|A^TA|$. $|A^T||A| = (-2)(-2) = 4$.

Practise with new numbers

Try it in Python

The program tests four properties on the matrix of the lesson: a row swap, multiplying a row, adding a multiple of a row, and the scalar multiple $2A$.

Common mistakes

• Writing $|kA| = k|A|$. For an $n \times n$ matrix it is $k^n|A|$.
• Multiplying the determinant when you add a multiple of a row. That operation changes nothing. Only multiplying a row changes the value.
• Forgetting the sign after a swap. Each swap of two rows changes the sign.
• Replacing a row by a multiple of itself plus another. If you replace $R_1$ by $2R_1 + R_2$, you have multiplied the row by $2$, so the value doubles.
• Writing $|A + B| = |A| + |B|$. This is false.

Remember these

• $|A^T| = |A|$, and $|AB| = |A||B|$.
• A swap changes the sign. Multiplying a row by $k$ multiplies $|A|$ by $k$. Adding a multiple of one row to another changes nothing.
• Equal or proportional rows, or a zero row, give $0$.
• $|kA| = k^n|A|$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A determinant has value $7$. Two rows are swapped. What is the new value?

2. Take out a common factor of row $1$ first, and then find this determinant: $

$.

= 3(10 - 12) = -6$.

3. Why is $

= 0$?

4. $A$ is $2 \times 2$ and $|A| = 5$. Find $|3A|$.

5. Use $|AB| = |A||B|$ to find $|AB|$ when $|A| = 3$ and $|B| = -4$.

References

• NCERT, Mathematics Textbook for Class XII, Part I, Chapter 4 (Determinants), the section on properties of determinants.
• Gilbert Strang, Introduction to Linear Algebra, the chapter on determinants (properties of the determinant).