A determinant is a word related to algebra, and it is found in most of the calculations in Mathematics. It is implemented in linear equations and also used in the computation of matrices.

Determinant Meaning

In linear algebra, the determinant is a scalar value that is computed from the elements of the square matrix that obeys certain properties of transformations which are defined by the matrix.

As we already know what a matrix is, it is an array of elements or numbers. Hence determinant math of a matrix is defined as the special number or value of a square matrix. Two vertical lines on either side are used to a denote a determinant as shown below-

|X| or det X-This denotes determinant of matrix X.

Suppose X is a matrix with its elements as shown,

\(\begin{bmatrix} 3 & 8\\ 4  & 6 \end{bmatrix}\)

The determinant of this matrix is |X| calculated as |X| = 6 * 3 – 8 * 4 = |-14|= +14. The determinant is applied to remove the negativity symbol from the values obtained on the calculation of matrices.

Determinants Properties

Some of the basic properties of determinants as listed below-

  • Transpose of a matrix- |Xt|= |X|: This property shows that the determinant for the matrix X is equal to the transpose of a matrix. For example- If |A| = 2 then the transpose matrix for the same will be |At|= -2
  • |X|= 0: In this property, the matrix consists of two equal lines having equal values. Hence the resultant matrix would always be zero.
  • |A-1| = 1/ A
  • |AB|= |A|. |B|: In this property, the det AB is equal to the product of the individual determinants.
  • If the elements of any two or two columns are identical, then the determinant value is equal to zero
  • If each element of a row or a column is multiplied by the constant, say X, then the value gets multiplied by the constant X
  • If any two rows and columns elements of a determinant are interchanged, then the sign of the determinant is also changed

Determinants Examples

Here you are provided with the examples to solve determinant math of 2 x 2 matrix and 3 x 3 matrix

How to Find the Determinant of 2 * 2 matrix?

Consider a matrix

A= \(\begin{bmatrix} a & b\\ c& d \end{bmatrix}\)

The det A for this matrix is calculated as follows-

|A| = ad-bc  is the scalar value

For example- if

A= \(\begin{bmatrix} 4 & 3\\ 6& 8 \end{bmatrix}\)

Then |A| = 4 * 8 – 6 * 3= 32- 18= 14.

How to Find the Determinant of 3 * 3 matrix?

Suppose A is a 3 * 3 matrix consisting of elements-

A= \(\begin{bmatrix} a & b & c\\ d& e& f\\ g& h& i \end{bmatrix}\)

Then the determinant of matrix A is calculated as

|A| = a(ei – fh) – b(di – fg) + c(dh – eg)

Here, an example is provided to find the determinant of 3 x 3 matrix

A = \(\begin{bmatrix} 6 & 1 & 1\\ 4& -2& 5\\ 2& 8& 7 \end{bmatrix}\)

The det A is given by |A| = [6×(-2×7 – 5×8) ]- [1×(4×7 – 5×2)] + [1×(4×8 – -2×2)] = [6×(-54)] – [1×(18) ]+ [1×(36)]

|A| = |-306| = 306.