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Determinants

Determinant of Matrices up to Order Three

Key Concepts

  • To every square matrix A = [aij] 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: MK is defined by f(A) = k, where AM and kK, 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| = = a11 a22a21 a12

  • 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| = kn |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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