Determinant of a Square Matrix

Basic Concept

Definition 1: The determinant, det A, also denoted |A|, of an n × n square matrix A is defined recursively as follows:

If A is a 1 × 1 matrix [a] (i.e. a scalar) then det A = a. Otherwise,

where A_ij is matrix A with row i and column j removed.

Note that if A = , then we use the notation  for det A.

Worksheet Functions

Excel Function: Excel provides the following function for calculating the determinant of a square matrix:

MDETERM(A): If A is a square array, then MDETERM(A) = det A. This is not an array function.

The Real Statistics function DET(A) provides equivalent functionality.

Properties

Property 1:

1. det A^T = det A
2. If A is a diagonal matrix, then det A = the product of the elements on the main diagonal of A

Proof: Both of these properties are a simple consequence of Definition 1

Property 2:  = ad  – bc

Example 1: Calculate det A where

From Definition 1 and Property 2, it follows that

Of course, we can get the same answer by using Excel’s function MDETERM(A).

Property 3: If A and B are square matrices of the same size then det AB = det A ∙ det B

Property 4: A square matrix A is invertible if and only if det A ≠ 0. If A is invertible then

The first assertion is equivalent to saying that a square matrix A is singular if and only if det A = 0.

Proof: Let A be an n × n matrix and define the adjunct of  to be the n × n matrix A^* = [c_ij] with c_ij = det A_ji where A_ji is as defined in Definition 1. Then it can be shown that AA^* = |A|I_n.

Now if |A| ≠ 0, then A^-1 = A^*/|A|, and so A is invertible. By Property 3, det A ⋅ det A^-1 = det AA^-1 = det I = 1, and so if A is invertible, then det A^-1 = 1/det A.

Conversely, suppose A is invertible and |A| = 0. By Property 3, 1 = det I = det AA^-1 = det A ⋅ det A^-1 = 0 ⋅ det A^-1 = 0, which is a contradiction, and so det A = |A| ≠ 0.

Property 5: Rules for evaluating determinants:

1. The determinant of a triangular matrix is the product of the entries on the diagonal.
2. When we interchange two rows, the determinant of the new matrix is the negative of the old one.
3. If we multiply one row by a constant, the determinant of the new matrix is the determinant of the old one multiplied by the constant.
4. If we add one row to another one multiplied by a constant, the determinant of the new matrix is the same as the old one.

Calculating the determinant

The rules in Property 5 are sufficient to calculate the determinant of any square matrix. The idea is to transform the original matrix into a triangular matrix and then use rule 1 to calculate the value of the determinant.

We now present an algorithm based on Property 5 for calculating det A, where A = [a_ij] is an n ×  n matrix. Start by setting the value of the determinant to 1, and then perform steps 1 to n as follows.

Step k – part 1(a): If a_kk ≠ 0, multiply the current value of the determinant by a_kk and then divide all the entries in row k by a_kk (rule 3 of Property 5).

Step k – part 1(b): If a_kk = 0, exchange row k with any row m below it (i.e. k < m ≤ n) for which a_mk ≠ 0, multiply the current value of the determinant by -1 (rule 2) and then perform step 1(a) above. If no such row exists then terminate the algorithm and return the value of 0 for the determinant.

Step k – part 2: For every row m below row k, add –a_mk times row k to row m (rule 4). This guarantees that a_ij = 0 for all i > k and j ≤ k.

After the completion of step n, we will have a triangular matrix whose diagonal contains all 1s, and so by rule 1, the determinant is equal to the current value of the determinant.

Example

Example 2: Using Property 5, find

We present the steps, proceeding from left to right and then from top to bottom in Figure 1. For each step, the rule used is specified as well as the multiplier of the determinant calculated up to that point.

Figure 1 – Calculating the determinant in Example 2

This shows that the determinant is -5, the same answer given when using Excel’s MDETERM function.

Observation: In step k – part 1(b) of the above procedure we exchange two rows if a_kk = 0. Given that we need to deal with roundoff errors, what happens if a_kk is small but not quite zero? In order to reduce the impact of round-off errors, we should modify step k – part 1 as follows:

Step k – part 1: Find m ≥ k such that the absolute value of a_mk is largest. If this a_mk ≈ 0 (i.e. |a_mk|< ϵ where ϵ is some predefined small value) then terminate the procedure. If m > k then exchange rows m and k.

Solving Systems of Linear Equations

The determinant can be used to solve systems of linear equations as described in Systems of Linear Equations via Cramer’s Rule. Also, the Gaussian elimination technique used to calculate the determinant can also be used to solve systems of linear equations.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Wikipedia (2012) Determinant
https://en.wikipedia.org/wiki/Determinant#:~:text=In%20mathematics%2C%20the%20determinant%20is,map%20represented%20by%20the%20matrix.