Determinants

What is Determinant of a Matrix?

Determinant of a matrix is a number that we can calculate from a square matrix.

Determinant is denoted by using two vertical lines, i.e. | |.

For example, if A is a square matrix, then its determinant will be denoted as |A|.

We use determinants in various fields of mathematics, for example:

• To find the inverse of a matrix.
• To solve two linear equations of two variables.
• In Calculus, etc.

So, now let’s understand how to find the determinant of a matrix.

Finding Determinant of a Matrix

We can find determinant of only square matrices, i.e. matrices having as many rows as columns.

There are various methods to find determinant of a matrix. But we generally use the Laplace expansion method.

It is easy to remember and apply on small matrices, such as a 2 × 2 matrix, or a 3 × 3 matrix etc. However, it becomes tedious to apply on bigger matrices.

Laplace expansion method

For 2 × 2 matrix

A 2 × 2 matrix has 2 rows and 2 columns, as the matrix displayed below:

$\begin{bmatrix} a & b \newline c & d \end{bmatrix}$

Its determinant, |A| = ad – bc

It can also be visualized via a picture:

For example, if B = $\begin{vmatrix} 1 & 2 \newline 3 & 4 \end{vmatrix}$ , then:

|B| = 1 × 4 – 2 × 3 = 4 – 6 = -2

For 3 × 3 matrix

A 3 × 3 matrix has 3 rows and 3 columns, as the matrix displayed below:

$\begin{bmatrix} a & b & c \newline d & e & f \newline g & h & i \end{bmatrix}$

Its determinant, |A| = a . $\begin{vmatrix} e & f \newline h & i \end{vmatrix}$ - b . $\begin{vmatrix} d & f \newline g & i \end{vmatrix}$ + c . $\begin{vmatrix} d & e \newline g & h \end{vmatrix}$ = a (ei - hf) - b (di - gf) + c (dh - ge)

So, we multiply ‘a’ with a determinant of a 2 × 2 matrix, which is formed by removing the elements in a’s row and column. We do the same for ‘b’ and ‘c’. In case of ‘b’, we also place a minus sign in front of it.

It can also be visualized via a picture:

For example, if C = $\begin{bmatrix} 1 & 2 & 3 \newline 4 & 5 & 6 \newline 7 & 8 & 9 \end{bmatrix}$ , then:

|C| = 1 (5 × 9 - 6 × 8) - 2 (4 × 9 - 6 × 7) + 3 (4 × 8 - 5 × 7) = 1 (45 - 48) - 2 (36 - 42) + 3 (32 - 35) = -3 - 2 (-6) + 3 (-3) = -3 + 12 - 9 = 0