Here you will learn how to find the determinant of matrix 2ร2 with examples.
Letโs begin โ
Determinant of Matrix 2ร2
If A = \(\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\) is a square matrix of 2ร2,
then \(a_{11}a_{22} โ a_{12}a_{21}\) is called the determinant of A.
i.e. | A | = \(\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}\)
= \(a_{11}a_{22} โ a_{12}a_{21}\)
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 :ย Let | A | = \(\begin{vmatrix} 5 & 4 \\ -2 & 3 \end{vmatrix}\)
By definition, we obtain
| A | = ( \(5\times 3\)) โ (\(4\times -2\)) = 15 + 8 = 23
Example 2 : find the determinant of \(\begin{vmatrix} sinx & cosx \\ -cosx & sinx \end{vmatrix}\).
Solution :ย ย Let | A | = \(\begin{vmatrix} sinx & cosx \\ -cosx & sinx \end{vmatrix}\)
By definition, we obtain
| A | = ( \(sin^2x\)) โ (\(-cos^2x\)) = \(sin^2x\) + \(cos^2x\) = 1
Example 3 : find the determinant of \(\begin{vmatrix} x โ 1 & 1 \\ x^3 & x^2 + x + 1 \end{vmatrix}\).
Solution : Let | A | = \(\begin{vmatrix} x โ 1 & 1 \\ x^3 & x^2 + x + 1 \end{vmatrix}\)
By definition, we obtain
| A |ย = (x โ 1)( \(x^2 + x + 1\)) โ (\(x^3\))
= \(x^3 โ 1\) โ \(x^3\) = -1
Example 4 : find the determinant of \(\begin{vmatrix} x^2 + xy + y^2 & x + y \\ x^2 โ xy + y^2 & x โ y \end{vmatrix}\).
Solution : Let | A | = \(\begin{vmatrix} x^2 + xy + y^2 & x + y \\ x^2 โ xy + y^2 & x โ y \end{vmatrix}\)
By definition, we obtain
| A |ย = ( \(x^2 + xy + y^2\))(x โ y) โ (\( x^2 โ xy + y^2\))(x + y)
= (\(x^3 โ y^3\)) โ (\(x^3 + y^3\)) = \(-2y^3\)
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 |ย = 1 โ ( \(log_ab \times log_ba\)) = 1 โ 1 = 0
