Determinants
Determinant of Matrices up to Order Three
Key Concepts

To every square matrix A = [a_{ij}] of order n, we can associate a number (real or complex) called the determinant of the square matrix A.

The determinant of a matrix A is denoted by A or det A or Δ.


If M is the set of square matrices, K is the set of numbers (real or complex) and f: M → K is defined by f(A) = k, where A∈M and k∈K, then f(A) is called the determinant of A.
Calculation of Determinants

Let A = [a] be the matrix of order 1. Accordingly, the determinant of A is defined to be equal to a.

Let be a matrix of order 2 × 2. Accordingly, the determinant of A is defined as det A = A = = a_{11} a_{22} − a_{21} a_{12}

To understand how to find the determinant of a 2 × 2 matrix with the help of an example, let us have a look at the following video.

To understand how to find the determinant of a 3 × 3 matrix, let us look at the following video.

The determinant of a matrix A can be obtained by expanding along any other row or any other column also.

For easier calculation, we expand the determinant along the row or column that contains the maximum number of zeroes.


If A = kB, where A and B are square matrices of order n, then A = k^{n} B, where n = 1, 2, 3.
Solved Examples
Example 1
Evaluate .
Solution:
= −5 (1 × 3 − 5 × 6) − 4 (3 × 3 − 2 × 6) + 2 (3 × 5 − 2 × 1)
= −5 (3 − 30) − 4 (9 − 12) + 2(15 − 2)
= −5(−27) − 4(−3) + 2(13)
= 135 + 12 + 26
= ...
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