Here you will learn what is determinant of matrix and formula for how to find the determinant of matrix of different order.
Let’s begin –
What is Determinant ?
If the equations \(a_1x + b_1\) = 0, \(a_2x + b_2\) = 0 are satisfied by the same value of x, then \(a_1b_2 – a_2b_1\) = 0.
The expression \(a_1b_2 – a_2b_1\) is called a determinant of the second order, and it is denoted by
\(\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}\)
A determinant of second order consists of two rows and two columns.
Next consider the system of equations \(a_1x + b_1y + c_1\) = 0, \(a_2x + b_2y + c_2\) = 0, \(a_3x + b_3y + c_3\) = 0
If these equations are satisfied by the same values of x and y, then on eliminating x and y we get,
\(a_1(b_2c_3 – b_3c_2)\) + \(b_1(c_2a_3 – c_3a_2)\) + \(c_1(a_2b_3 – a_3b_2)\) = 0
The expression on the left is called a determinant of the third order, and is denoted by
\(\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}\)
How to Find the Determinant of Matrix
Determinant of Matrix of Order 1
If A = \([a_1]\) is a square matrix of order 1, then the determinant of A is defined as
| A | = \(a_1\) or, \(|a_1|\) = \(a_1\)
Determinant of Matrix of Order 2
If A = \(\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}\) is a square matrix of order 2,
then the expression \(a_1b_2 – a_2b_1\) is defined as the determinant of A.
i.e. | A | = \(\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}\) = \(a_1b_2 – a_2b_1\)
Determinant of Matrix of Order 3
If A = \(\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\) is a square matrix of order 3,
then the expression \(a_1(b_2c_3 – b_3c_2)\) – \(b_1(a_2c_3 – a_3c_2)\) + \(c_1(a_2b_3 – a_3b_2)\) is defined as the determinant of A.
i.e. | A | = \(\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}\)
= \(a_1\begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix}\) – \(b_1\begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix}\) + \(c_1\begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix}\)
= \(a_1(b_2c_3 – b_3c_2)\) – \(b_1(a_2c_3 – a_3c_2)\) + \(c_1(a_2b_3 – a_3b_2)\)
Related Questions
Find the determinant of \(\begin{vmatrix} sinx & cosx \\ -cosx & sinx \end{vmatrix}\).
Find the determinant of A = \(\begin{bmatrix} 3 & -2 & 4 \\ 1 & 2 & 1 \\ 0 & 1 & -1 \end{bmatrix}\).
