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.

Facts:

• A determinant is applicable only for square matrix (for eg.1×1, 2×2, 3×3, 4×4)
• A determinant can be real or complex number
• |A| does not express the modulus of A here, but the determinant of matrix A
• If all the elements of 2×2 matrix are the same, then the determinant will be zero
• If all the entries of a row or a column in a 2×2 matrix are zero, then the determinant is also zero
• The determinant of the product of two matrices is equal to the product of their determinants, respectively. |AB| = |A| |B|.

The determinant of a matrix of order 2, is denoted by A = [aij]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;

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

Then the determinant of A is defined as:

Det (A) = $|A| = \begin{vmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{vmatrix}$

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 $A=\begin{bmatrix} 3 & 5\\ 9 & 8 \end{bmatrix}$

Solution: Given,

$A=\begin{bmatrix} 3 & 5\\ 9 & 8 \end{bmatrix}$

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

Det A = $|A| = \begin{vmatrix} 3 & 5\\ 9 & 8 \end{vmatrix}$

|A| = 3 x 8 – 5 x 9

|A| = 24 – 45

|A| = -21

Q.2: Find the determinant of 2×2 matrix $A=\begin{bmatrix} 0 &1 \\ 4 &1 \end{bmatrix}$.

Solution:

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

$|A| = \begin{vmatrix} 0 & 1\\ 4 & 1 \end{vmatrix}$

|A| = 0.1 – 1.4

|A\ = -4

Q.3: If $A=\begin{bmatrix} 6&9 \\ -4&7 \end{bmatrix}$, then find the value of det (A).

Solution: Given,

$A=\begin{bmatrix} 6&9 \\ -4&7 \end{bmatrix}$

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 $A=\begin{bmatrix} 1&2 \\ 1&1 \end{bmatrix}$ and $B=\begin{bmatrix} 2&1 \\ 1&1 \end{bmatrix}$ are two matrices, then find the determinant of the product of A and B. Also check if |AB| = |A|.|B|.

Solution: Given,

$A=\begin{bmatrix} 1&2 \\ 1&1 \end{bmatrix}$ $B=\begin{bmatrix} 2&1 \\ 1&1 \end{bmatrix}$

Product of matrices A and B:

$A.B = \begin{bmatrix} 1.2+2.1&1.1+2.1 \\ 1.2+1.1&1.1+1.1 \end{bmatrix}$ $A.B = \begin{bmatrix} 4&3 \\ 3&2 \end{bmatrix}$

Now, determinant of A.B, will be;

$|A.B| = \begin{vmatrix} 4&3 \\ 3&2 \end{vmatrix}$

|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.

$|A| = \begin{vmatrix} 1&2\\ 1&1 \end{vmatrix}$

|A| = 1.1 – 2.1 = -1

$|B| = \begin{vmatrix} 2&1\\ 1&1 \end{vmatrix}$

|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|

Practice Questions

Find the determinant of matrices of order two, given below:

1. $|A|=\begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix}$
2. $|B| = \begin{bmatrix} 21&22\\ 11&12 \end{bmatrix}$

How to find the determinant of a matrix of order 2?

Cross multiply the elements of the 2×2 matrix and then find their difference. Hence, the determinant of the matrix is calculated.

If matrix A has elements a11, a12, a21 and a22, then find the determinant of A.

The determinant of matrix A will be:

|A| = a11.a22 – a12.a21

What is a second order determinant?

The determinant of a 2×2 matrix or order two matrix, is called second order determinant.

What is the determinant of a 2×1 matrix?

We cannot determine the determinant of a 2 x 1 matrix. To find the determinant, the matrix should be a square matrix.