# Determinant of Matrix 2×2 with Examples

Here you will learn how to find the determinant of matrix 2×2 with examples.

Let’s begin –

## Determinant of Matrix 2×2

If A = $$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$ is a square matrix of 2×2,

then $$a_{11}a_{22} – a_{12}a_{21}$$ is called the determinant of A.

i.e. | A | = $$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}$$

= $$a_{11}a_{22} – a_{12}a_{21}$$

Thus, the determinant of a square matrix of order 2 is equal to the product of the diagonal elements minus the product of off-diagonal elements.

Example 1 : find the determinant of $$\begin{vmatrix} 5 & 4 \\ -2 & 3 \end{vmatrix}$$.

Solution : Let | A | = $$\begin{vmatrix} 5 & 4 \\ -2 & 3 \end{vmatrix}$$

By definition, we obtain

| A | = ( $$5\times 3$$) – ($$4\times -2$$) = 15 + 8 = 23

Example 2 : find the determinant of $$\begin{vmatrix} sinx & cosx \\ -cosx & sinx \end{vmatrix}$$.

Solution :  Let | A | = $$\begin{vmatrix} sinx & cosx \\ -cosx & sinx \end{vmatrix}$$

By definition, we obtain

| A | = ( $$sin^2x$$) – ($$-cos^2x$$) = $$sin^2x$$ + $$cos^2x$$ = 1

Example 3 : find the determinant of $$\begin{vmatrix} x – 1 & 1 \\ x^3 & x^2 + x + 1 \end{vmatrix}$$.

Solution : Let | A | = $$\begin{vmatrix} x – 1 & 1 \\ x^3 & x^2 + x + 1 \end{vmatrix}$$

By definition, we obtain

| A |  = (x – 1)( $$x^2 + x + 1$$) – ($$x^3$$)

= $$x^3 – 1$$ – $$x^3$$ = -1

Example 4 : find the determinant of $$\begin{vmatrix} x^2 + xy + y^2 & x + y \\ x^2 – xy + y^2 & x – y \end{vmatrix}$$.

Solution : Let | A | = $$\begin{vmatrix} x^2 + xy + y^2 & x + y \\ x^2 – xy + y^2 & x – y \end{vmatrix}$$

By definition, we obtain

| A |  = ( $$x^2 + xy + y^2$$)(x – y) – ($$x^2 – xy + y^2$$)(x + y)

= ($$x^3 – y^3$$) – ($$x^3 + y^3$$) = $$-2y^3$$

Example 5 : find the determinant of $$\begin{vmatrix} 1 & log_ba \\ log_ab & 1 \end{vmatrix}$$.

Solution : Let | A | = $$\begin{vmatrix} 1 & log_ab \\ log_ab & 1 \end{vmatrix}$$

By definition, we obtain

| A |  = 1 – ( $$log_ab \times log_ba$$) = 1 – 1 = 0