Determinant of a Matrix of Order Two

Determinants of a matrix of order two can be evaluated for a square matrix of dimensions 2 x 2. To determine the determinant of a 2×2 matrix, we have to find the difference of cross multiplication of the elements. Therefore, we don’t have to use the calculator here to find the determinant of order 2 matrix, quickly. Determinant is calculated only for a square matrix.

The determinant of a matrix of order 2, is denoted by A = [a_ij]_2×2, where A is a matrix, a represents the elements i and j denotes the rows and columns, respectively. Let us learn more about the determinant formula for a 2 x 2 matrix along with examples to understand better.

Determinant of a 2 x 2 Matrix

Suppose, A = [aij] is a 2 x 2 matrix (order two matrix), such that;

Then the determinant of A is defined as:

Det (A) =

Det (A) = a₁₁.a₂₂ – a₁₂.a₂₁

Or

|A| = a₁₁.a₂₂ – a₁₂.a₂₁

This is the determinant formula for matrix of order two.

Solved Examples

Q.1: Find the determinant of matrix

Solution: Given,

By the determinant formula, determinant of matrix A, is:

Det A =

|A| = 3 x 8 – 5 x 9

|A| = 24 – 45

|A| = -21

Q.2: Find the determinant of 2×2 matrix

Solution:

The determinant of a matrix A of order two will be:

|A| = 0.1 – 1.4

|A\ = -4

Q.3: If

Solution: Given,

Determinant of matrix of order two is given by:

Det (A) = a₁₁.a₂₂ – a₁₂.a₂₁

In the given matrix A,

a₁₁ = 6

a₁₂ = 9

a₂₁ = -4

a₂₂ = 7

Hence,

|A|=\begin{vmatrix}

6&9 \\

-4&7

\end{vmatrix}

|A| = 6 x 7 – 9 x (-4)

= 42 + 36

= 78

Thus, the required determinant is 78.

Q.4: If

Solution: Given,

Product of matrices A and B:

Now, determinant of A.B, will be;

|A.B| = 4.2 – 3.3

|A.B| = 8 – 9

|A.B| = -1 ………………..(i)

Now, we need to check if |A.B| = |A|.|B|

Let us find the determinant of matrices A and B separately.

|A| = 1.1 – 2.1 = -1

|B| = 2.1 – 1.1 = 1

So,

|A|.|B| = (-1)x(1) = -1 ……….(ii)

Therefore, by (i) and (ii), it is proved that;

|A.B| = |A|.|B|