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

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